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First published 1987

Printed in Great Britain at the University Press, Cambridge
ISBN 82 00 18386 6 paperback (Scandinavia only)

British Library cataloguing in publication data

Taylor, Michael, 1942-

The possibility of cooperation. — [Rev.ed.]

— (Studies in rationality and social change)

1. Anarchism 2. State, The

3. Prisoners' dilemma game

I. Title II. Taylor, Michael, 1942-

Anarchy and cooperation III. Series

320.1'01 HX833

Library of Congress cataloguing in publication data

Taylor, Michael

The possibility of cooperation.

(Studies in rationality and social change)

Rev. ed. of: Anarchy and cooperation. cl976.

Bibliography.

Includes index.

1. Anarchism. I. Taylor, Michael

Anarchy and cooperation. II. Title. III. Series.

HX833.T38 1987 320.5'7 86-30969

ISBN 521 32793 8 hard covers
ISBN 521 33990 1 paperback

Contents

Preface ix

1 Introduction : the problem of collective action 1
The provision of public and other non-excludable goods 5
The Prisoners' Dilemma 13
The problem of collective action 18
Time and the lone exploiter 20
Solutions? Community, states, entrepreneurs, property

rights and norms 21

Plan of the rest of the book 30

2 The Prisoners' Dilemma, Chicken and other

games in the provision of public goods 34

Alternatives to the Prisoners' Dilemma 35

An JV-person game of Chicken 40

Mutual aid, fisheries and voting in committees 43
Pre-commitment as a risky decision and the prospects for

cooperation in Chicken games 45

Continuous strategy sets 49

Cournot analysis 56

A summary remark 58

3 The two-person Prisoners' Dilemma supergame 60
Supergames 60
Unconditional Cooperation and Defection 64
The possibility of conditional Cooperation 65
An Assurance game 67
Conditions for (B, B) to be an equilibrium 67
Axelrod's tournaments 69
Coordination equilibria 71
Other mutual Cooperation equilibria 73
Taking it in turns to Cooperate 76
Outcomes 78

CONTENTS

4 The iV-person Prisoners' Dilemma supergame
Payoffs in the constituent game

Unconditional Cooperation and Defection
Conditional Cooperation
Subgroups of Cooperators

Chickens nesting in the Prisoners' Dilemma supergame
Other Cooperative equilibria
An example

Alternation between blocks of conditional cooperators
Summary and discussion of results
A more realistic model

5 Altruism and superiority
Altruism in two-person games
An N-person Game of Difference

6 The state
Hobbes's Leviathan
Hume's Leviathan

7 Epilogue: cooperation, the state and anarchy
International anarchy

The destruction of community
The decay of voluntary cooperation
Rationality

Annex : the theory of metagames

Notes

Bibliography
Index

This book is a revised edition of
the author's Anarchy and Cooperation
first published in 1976 by
John Wiley and Sons Ltd
and now out of print

Preface

The Possibility of Cooperation is, I hope, much more than the preceding
statement might suggest, 'a new edition of Anarchy and Cooperation'.
The chapter dealing with the political theories of Hobbes and Hume is
the only one to survive intact, though the short chapter on altruism and
the rather informal Epilogue contain much of the corresponding
chapters of the earlier book. All the rest has been substantially re-cast.
Amongst other things, I have devoted much more of the book to the
theory of collective action, which in the years since writing Anarchy and
Cooperation I have come to see as absolutely fundamental to the study of
politics and history, and in particular I have devoted more space to the
theory of conditional cooperation in supergames and at the same time
tried to make the formal treatment of this subject more accessible to the
non-mathematical reader than it was before.

Anarchy and Cooperation was constructed as a critique of the liberal
theory of the state, according to which the state is necessary because
people, being rational, will not voluntarily cooperate to provide
themselves with public goods, in particular the basic public goods of
social order and defence. At the heart of this critique was a study of the
Prisoners' Dilemma supergame, for it had been (as it still is) widely
assumed that the problem of public goods provision, or 'the collective
action problem' more generally, has the form of a Prisoners' Dilemma.
So in the central chapters of Anarchy and Cooperation I wanted to
question whether the Prisoners' Dilemma game was the correct
representation of preferences in public goods interactions and then,
assuming that it was, to examine the prospects for voluntary cooper-
ation in this game. At that time there had been little theoretical study of
the iterated Prisoners' Dilemma, especially in its JV-person form. Yet it

PREFACE

seemed obvious to me that the one-shot game had little to do with
anything of importance in the real world and that most problems of
public goods interaction and of collective action generally had more
than two players. Accordingly, I devoted the central chapter of Anarchy
and Cooperation to an analysis of the two-person and iV-person
Prisoners' Dilemma supergames (in which the basic games are played an
indefinite number of times).

This purely formal treatment of collective action and the Prisoners'
Dilemma supergame could be - and generally was - read and used
independently of the critique of the liberal theory of the state. In this new
edition, the critique stands, though parts of the old final chapter
('Anarchy') have been rewritten and are offered less confidently in the
Epilogue. (Further doubts about 'anarchy' have also crept into my
Community, Anarchy and Liberty, which is in some respects a companion
piece to this book, and some day I hope to publish a third volume giving
a fuller and more historical account of the state.) But, as before, the
analysis of collective action and the Prisoners' Dilemma supergame can
be read - and assessed - independently of the critique, and this part of the
book has been substantially recast and extended. Chapter 3 of the old
book - the central chapter dealing with the Prisoners' Dilemma
supergame - has been almost completely rewritten: the material has
been re-organized, some of the arguments have been modified and ex-
tended, and several new sections have been added. I have endeavoured
to make the analysis more accessible to readers with little training in
mathematics, and in pursuing this end I found that I could derive the
same results - and indeed strengthen and extend them - using far less
algebra. The new treatment is split between two chapters. To the first of
these, dealing with the two-person supergame, I have added amongst
other things a brief section commenting critically on related work by
Robert Axelrod and Russell Hardin published since Anarchy and
Cooperation appeared. All the results in this chapter which formerly gave
only necessary conditions for various equilibria now provide conditions
which are both necessary and sufficient. To the second of these chapters,
dealing with the iV-person supergame, I have added, partly in response
to a critical point raised by several readers of Anarchy and Cooperation, a
section which shows how this game can give rise to a Chicken game (thus
introducing a troublesome indeterminacy into the analysis of the N-
person case).

PREFACE

The brief account of the effects of altruism on behaviour in Prisoners'
Dilemma games has also been pruned of some unnecessary algebra.
Removing much of the mathematics from this chapter and from the
treatment of the Prisoners' Dilemma supergame has made redundant
the old chapter 5, which provided an informal summary of the
mathematical parts of the book, and I have therefore removed it.

Chapter 2 is almost entirely new. Amongst other things it develops an
argument to the effect that in public goods problems individual
preferences at any point in time are not necessarily those of a Prisoners'
Dilemma; that in fact many interesting public goods problems, both
two-person and N-person, are better represented by Assurance and
Chicken games; and that where individuals can choose from a continu-
ous range of strategies their preferences are quite likely to be those of a
Chicken-like game or of a hybrid between a Chicken and an Assurance
game. In all of these cases, arguably, some cooperation is more likely
than in the case of a Prisoners' Dilemma, even when the game is played
only once. Chicken emerges from this discussion, and from the account
in chapter 4 of the iV-person Prisoners' Dilemma supergame, as an
important game. A little analysis of this neglected game is offered here,
but much more needs to be done.

Much of chapter 1 is also new (i.e., not to be found in Anarchy and
Cooperation). Partly in order to make the book a little more self-
contained, I have given in the opening section a fuller summary of
Olson's 'logic of collective action' and in particular his important
argument that the larger a group is, the less likely are its members to
cooperate voluntarily in the provision of a public good. I agree with
those writers who have shown that this 'size' effect is not always present,
but I want also to argue that not too much weight should be attached to
any of these 'size' arguments - Olson's or his critics' - because the model
from which they derive, being entirely static, is an unrealistic represen-
tation of almost all problems of public goods provision, which are of
course typically recurring. In a dynamical formulation, but not of course
in a static one, there is the possibility of conditional cooperation; and I
would argue that the 'size' effect which should be taken most seriously is
the increased difficulty of conditional cooperation as group size
increases.

Many of the arguments in this book apply not just to problems of
public goods provision but to 'collective action problems', a much larger

xii

PREFACE

category. So I added to chapter 1 a brief attempt to characterize this
category. I have also added a new section which both characterizes the
range of possible solutions to collective action problems and examines
the particular claims that such problems can be solved by political
entrepreneurs, by the establishment of property rights and by norms.

I have made no attempt to provide a comprehensive survey of
developments in the study of public goods, collective action and
supergames that have occurred since the publication of Anarchy and
Cooperation. Although I have commented briefly on the most closely
related work (such as that of Hardin and Axelrod) and have made a
number of excisions and revisions that were prompted by some of these
developments, I have not, for example, reviewed either the extensive
recent theoretical work of economists and game theorists on supergames
or the even more extensive experimental work of psychologists,
sociologists and others on iterated Prisoners' Dilemmas. I still believe
that, if problems of public goods provision or other collective action
problems are to be modelled as iterated games, then the appropriate
model is a game iterated an indefinite number of times in which players
discount future payoffs. But although most economists seem to share
this view, most of the recent mathematical work on supergames treats
either finitely iterated games or infinite supergames without discounting.
I have, however, provided some references to this work for those wishing
to pursue it. As for the vast experimental literature, I remain un-
persuaded that its results can tell us much about the real world beyond
the experiments. There are two general problems with this literature.
First, the experiments are of short duration relative to the time span of
the processes in the real world they are supposed to simulate; so
discounting plays no role in the experiments, while in the real world it is
crucial. Second, the experimental payoffs are generally too small to elicit
rational behaviour. It is therefore not surprising that these experiments
yield such mixed results, with some studies finding significant free riding
and 'size' effects and others not.

Much more promising, in my view, are historical studies of collective
action. I have not tried to summarize such studies, which are thin on the
ground, for the best of them are still in progress and yet to be published,
but a good example of the sort of thing I have in mind is John Bowman's
forthcoming study of collective action amongst capitalists (based on his

PREFACE

xiii

doctoral dissertation, 'Economic competition and collective action : the
politics of market organization in the bituminous coal industry,
1880-1940', University of Chicago, 1984), which amongst other things
uses the theory of conditional cooperation in Prisoners' Dilemma
supergames to scrutinize the historical data. Arduous though it is
compared with experimental work, this, it seems to me, is the kind of
empirical work we need more of.

Anarchy and Cooperation owed much to the kind help of Brian Barry,
Alan Carling, Ian Grant and Michael Nicholson, who commented
extensively on the manuscript. If I had heeded more of Brian Barry's
advice at that time, much of the rewriting that has gone into this new
version would not have been necessary. For their help in various ways in
the preparation of this edition I should like also to thank Jon Elster, who
suggested it; Russell Hardin, who in conversation has helped me to
clarify a number of points; Dawn Rossit, who as my research assistant in
Seattle in the autumn of 1985 gave me valuable help in connection with
the discussion of property rights in chapter 1 ; and Hugh Ward, who
collaborated with me in work on the game of Chicken on which a part of
chapter 2 is based. The first edition was written during 1973-74 when I
was a Fellow at the Netherlands Institute for Advanced Study, whose
staff I would like to thank for their hospitality and help. For freedom to
work on this new edition, I am grateful to the University of Essex, and for
provision of a research assistant and for general support I would like
to thank the chairman of the Department of Political Science at the
University of Washington in Seattle.

M.J.T.

1. Introduction: the problem of collective
action

The most persuasive justification of the state is founded on the argument
that, without it, people would not successfully cooperate in realizing
their common interests and in particular would not provide themselves
with certain public goods : goods, that is to say, which any member of the
public may benefit from, whether or not he or she contributes in any way
to their provision. The most appealing version of this justification would
confine the argument about voluntary cooperation to what are sup-
posed to be the most fundamental public goods: goods (or services)
which are thought to be preconditions of the pursuit and attainment of
all other valued ends, including less basic public goods, and are therefore
desired by everyone within the jurisdiction of the state in question.

The Possibility of Cooperation is a critique of this justification of the
state, and the heart of the critique (chapters 2-4 below) is a detailed
study of cooperation in the absence of the state and of other kinds of
coercion. (The arguments about public goods provision and the theory
of cooperation which make up these chapters can be read - and assessed
- independently of the critique of the Hobbesian justification of the
state.)

Hobbes's Leviathan was the first full expression of this way of
justifying the state. The public goods with which he was principally
concerned were social order - domestic peace and security - and defence
against foreign aggression. Without these, very little else that was worth
having could be had. Without internal and external security, there would
be not only actual violence but such pervasive uncertainty as to
undermine the incentive to invest resources in any projects with delayed
returns. But although everyone would prefer the condition of peace and
security that mutual restraint ensures to the 'war of all against all' that is
the result of everyone pursuing his own interests without restraint, no

2 introduction: the problem of collective action

individual has the incentive, in the absence of the state, to restrain
himself. It is therefore rational, says Hobbes, for everyone to institute a
government with sufficient power to ensure that everybody keeps the
peace. 1

Many writers who came after Hobbes, including some who professed
no sympathy with what they took to be Hobbes's ideas, have taken over
the core of his case for the state. Most economists who nowadays write
about public goods believe that the failure of people to provide
themselves voluntarily with these goods constitutes at least a prima facie
case for state activity, and most of them presume that the state is the only
means for remedying this failure. 2 (For nearly all the rest, the remedy is
to establish or extend private property rights. I'll comment briefly on this
view later in the chapter.)

Of course, many people believe that the state can be justified on
further grounds and that it has functions other than that of providing
public goods. Certainly, modern states do more than provide such
goods. However, the justification I wish to criticize here is common to
the arguments of nearly all those who believe that the state is necessary.
Its persuasiveness lies in the fact that the state, on this view, exists to
further common interests, to do what everybody wants done. Other
arguments - for example, that income redistribution is desirable and can
be brought about only through the intervention of the state - do not
appeal to common interests, not, at any rate, in an obvious or
uncontroversial way.

In recent years, this argument about the necessity of the state has
found new supporters amongst those concerned with the degradation of
the environment, the depletion of non-renewable resources and rapid
population growth. According to them, people will not voluntarily
refrain from discharging untreated wastes into rivers and lakes, from
hunting whales and other species threatened with extinction, from
having 'too many children', and so on. Only powerful state action, they
say, can solve or avert these problems, which are the consequence of
failures to provide public goods (and more generally 'non-excludable'
goods - about which more later). For many environmentalists, some of
these public goods are at least as fundamental as peace and security were
for Hobbes. Continued failure to provide them will eventually result in
ecological catastrophe. Without them, the life of man will not just be
'solitary, poore, nasty, brutish, and short'; it will be impossible. 3

introduction: the problem of collective action 3

There have of course been other responses to the environmental crisis.
In particular, some writers, who probably did not think of themselves as
anarchists, have come to embrace essentially communitarian anarchist
ideas. However, of those who desire on ecological grounds the goal of a
social organization along communitarian anarchist lines, there are very
few who believe that a transition to such a society can be made without
extensive state activity. 4

As for the members of governments themselves, and indeed of most
political parties, especially in industrialized countries, they generally do
not recognize that there is or will be an environmental crisis and they
believe, not unnaturally, that pollution and resource depletion are
problems which can be adequately dealt with by minor modifications
within the present institutional framework of whatever country they
happen to live in. 5 Generally speaking, the proposed modifications
involve an extension of state activity, in the form of state-enforced
pollution standards and resource depletion quotas, taxes on industrial
pollution, government subsidies and tax credits for the development of
pollution control technology, and so on. Most economists who have
written on problems of pollution and resource depletion have also
confined their discussions to 'solutions' of this sort, or otherwise have
recommended the extension of private property rights.

Much of what I shall have to say in this book will in fact apply, not just
to the voluntary provision of public goods but to 'collective action
problems', a much larger category. The defining characteristic of a
collective action problem, as I shall use this expression, is very roughly
that rational egoists are unlikely to succeed in cooperating to promote
their common interests. (I will clarify this in a later section.) On this
account, as we shall see, the category of collective action problems
includes many but not all public goods problems. There is, in particular,
a very important class of collective action problems which arise in
connection with the use of resources to which there is open access -
resources, that is, which nobody is prevented from using. These
resources need not be public goods, as I will define them shortly. Garrett
Hardin's well-known 'tragedy of the commons' concerns resources of
this kind. 6

Hardin asks us to imagine a common, a pasture open to all. The village
herdsmen keep animals on the common. Each herdsman is assumed to
seek to maximize his own gain. As long as the total number of animals is

4 introduction: the problem of collective action

below the carrying capacity of the common, a herdsman can add an
animal to his herd without affecting the amount of grazing of any of the
animals, including his own. But beyond this point, the 'tragedy of the
commons' is set in motion. Asking himself now whether he should add
another animal to his herd, he sees that this entails for him a gain and a
loss: on the one hand, he obtains the benefit from this animal's yield
(milk, meat or whatever); on the other hand, the yield of each of his
animals is reduced because there is now overgrazing. The benefit
obtained from the additional animal accrues entirely to the herdsman.
The effect of overgrazing, on the other hand, is shared by all the
herdsmen; every one of them suffers a slight loss. Thus, says Hardin, the
benefit to the herdsman who adds the animal is greater than his loss. He
therefore adds an animal to the common. For the same reason, he finds
that it pays him to add a second animal, and a third, a fourth and so on.
The same is true for each of the other herdsmen. The result is that the
herdsmen collectively bring about a situation in which each of them
derives less benefit from his herd than he did before the carrying capacity
of the common was exceeded. The process of adding animals may indeed
continue until the ability of the common to support livestock collapses
entirely.

For similar reasons, many species of fish and whales are hunted
without limit and in some cases brought close to extinction: the oceans
are like a great common. For similar reasons, too, lakes and rivers are
polluted, since each polluter finds that the costs of treating his wastes
before discharging them or of modifying his product are too great in
comparison with what he suffers from the decline in the quality of the air
or water caused by his effluent.

In all these situations, we can say that it is in every individual's interest
not to restrain himself (from adding animals to the common, polluting
the lake, etc.) but the result of everyone acting without restraint is a state
of affairs in which every individual is less well off than he would be if
everybody restrained themselves.

In such situations, we might expect people to make an agreement in
which they all promised to restrain themselves. However, in the absence
of the state (or some other form of coercion), no individual has any
greater incentive to abide by the agreement than he had to restrain
himself before the agreement was made.

I shall later question whether grazing commons - such as those which

introduction: the problem of collective action 5

were part of the European open-field systems or those which were once
widespread in pastoral economies - do in fact typically have open access.
But certainly there are many such resources to be found in other
contexts.

The recent history of the whale 'fisheries' provides a sad example.
During the 1950s and 1960s, unlimited killing of blue and fin-back
whales, which are the biggest, brought these two species close to
extinction. When stocks of blues and fin-backs became very low, the other
large species were hunted without limit. In each case the annual harvest
far exceeded the maximum sustainable yield, that is, the maximum
number which can be replaced each year through reproduction (and the
whale hunters knew this). The profitability of whaling declined, and
most of the former whaling countries were obliged one by one to leave
the industry (so that, by 1968, there were only two countries, Japan and
the USSR, left in the field). It seems fairly certain that if it were not for the
diminished profits from hunting a sparse population, the blue whale and
other species would in fact have been hunted to extinction. After the
Second World War, the International Whaling Commission was set up
by the seventeen countries who were then interested in whaling and was
charged with regulating harvests and ensuring the survival of threatened
species. Until very recently, this Commission, which has no powers of
enforcement, has not been very successful. Its members were often
unable to agree to impose the quotas recommended by biologists, or else
they could agree only to limits in excess of these recommendations; and
when the Commission did decide either to limit harvests or to protect a
species completely, the agreement was not always observed by every
country. 7

The provision of public and other non-excludable goods

Before embarking on a detailed analysis, we need to define a little more
carefully some of the terms that have already been used, and in particular
the notions of a 'public good' and a 'collective action problem'.

I shall say that a good or service is a public good (or collective good) if
it is in some degree indivisible and non-excludable. A good is said to
exhibit perfect indivisibility or jointness of supply (with respect to a given
set of individuals, or public) if, once produced, any given unit can be
made available to every member of the public, or equivalently if any

6 introduction: the problem of collective action

individual's consumption or use of the good does not reduce the amount
available to others. 8 A good is said to exhibit non-excludability (with
respect to some group) if it is impossible to prevent individual members
of the group from consuming it or if such exclusion is 'prohibitively
costly' (a notion whose precise definition matters for some purposes but
not for mine here).

A perfectly divisible good is one that can be divided between
individuals. Once any part of it is appropriated by any individual, the
same part cannot be made available to others; and once any unit of it is
consumed by any individual, the amount available for consumption by
others is reduced by the whole of that unit. A loaf of bread and a pot of
honey are examples of perfectly divisible goods. A good which is
perfectly divisible is called a private good. Thus, in order to be public, a
good must exhibit some degree of indivisibility or jointness. 9

A good may be indivisible yet excludable. A road or bridge or park can
be provided in this form. Once supplied to one individual, it can be made
available to others, but it need not be, for it is possible and may not be
prohibitively costly to exclude particular individuals. Hence tolls and
admission charges can be imposed. Goods like these can be provided in
an excludable or non-excludable mode. Indivisibility, then, does not
imply non-excludability. Furthermore, divisibility does not entail ex-
cludability, although important examples of non-excludable, divisible
goods are not easy to come by: economists have suggested such
examples as a garden of flowers, whose nectar can be appropriated by
individual bees but particular bees cannot be excluded from
consumption.

If an individual is not excluded from consumption or use of a public
good, it is possible for him to be a free rider on the efforts of others, that is,
he can consume or use the public good that is provided by others (unless
of course everyone else tries to free-ride as well!). Whether or not he will
in fact be a free rider is something we have to examine.

Free rider problems (and hence, as we shall see, collective action
problems) can arise where there is non-excludability but not indivis-
ibility. In fact non-excludability (or de facto non-exclusion) and
divisibility (at least in principle) characterize Garrett Hardin's 'com-
mons', or any resource to which there is open access, such as a common
fishing ground, a common underground reservoir of oil or water, or the
open range on the Great Plains before any property rights were

introduction: the problem of collective action 1

established (including the 'common property' rights that the cattlemen
tried to maintain when they formed associations to regulate access and
use). I will have a little more to say about such resources in a later
section. With some of them, exclusion is possible and economically
feasible, but whether or not there is in fact exclusion, consumption or use
by one individual reduces the amount available to others and any
cutting back on consumption by one individual allows others to
consume more.

Most, if not all, public goods interactions are characterized by a
certain degree of rivalness. It is normally said that a good is rival to the
extent that the consumption of a unit of the good by one individual
decreases the benefits to others who consume that same unit. Obviously,
in the case of a perfectly divisible good the consumption of a particular
unit prevents any other individual from consuming it at all, so that there
can be no question of his benefiting from consumption. In this case we
might say that the good is perfectly rival. But non-rivalness is not the
same thing as indivisibility, as some writers like to say, even though they
are usually closely associated. Where there is some degree of divisibility,
consumption reduces the amount available to others; but where there is
some degree of rivalness, consumption reduces the benefits to other
consumers. An individual's benefit from consumption may not change at
all as the amount available for consumption declines, until some
threshold of 'crowding' is reached. In fact, although others' consumption
usually lowers an individual's utility - as is normally the case with
congested parks, beaches and roads and with various forms of pollution
- some individuals' utilities may rise as the number of other consumers
increases, at least up to a point; they may, for example, prefer a semi-
crowded beach or park to an empty one. This brings out the point that
rivalness, unlike indivivisibility, is strictly speaking a property of
individuals (or of their utility functions), not of the goods themselves.

Rivalness is clearly important in the analysis of collective action
problems. As we would expect - and as we shall see in the next chapter -
it plays a crucial role in the analysis of 'size' effects. For just as the
alternatives actually available to an individual must change as the
number of people in the group increases if there is some degree of
divisibility, so the utilities of these alternatives must change with group
size if there is rivalness.

I said in an earlier section that social order and national defence are

8 introduction: the problem of collective action

public goods. This needs some qualifying and we are now in a position to
do so. National defence can in fact be decomposed (on a first rough cut)
into deterrence, which is a pure public good because it is both perfectly
indivisible and completely non-excludable, and protection from attack,
which is imperfectly indivisible and more or less excludable depending
on the form it takes. (And of course the production of the means to these
ends produces incidental private goods, including income for sharehol-
ders and employees of business firms.) 10 The security of persons and
their property which I have taken to be constitutive of social order is
similarly the product of a variety of goods and services, which range (in a
modern society) from purely private goods like locks and private
bodyguards through such services as police forces and law courts to
deterrence, which, again, can be purely indivisible - as it would be if the
fact that attack was deterred on one individual did not diminish the
deterrent effect with respect to other individuals. 1 1

Under what circumstances, then, will people cooperate to provide a
public good or any non-excludable good which the members of a group
have a common interest in providing? The now standard answer to this
question (which, however, needs to be qualified, as we shall see) is the one
provided by Mancur Olson in his well-known study, The Logic of
Collective Action. Olson's main contention is that 'the larger a group is,
the farther it will fall short of providing an optimal supply of any
collective good, and the less likely that it will act to obtain even a
minimal amount of such a good. In short, the larger the group, the less
likely it will further its common interests.' 12

There are three arguments in support of this conclusion to be found in
Olson's book. Before setting them out, we need some definitions. A
group is privileged if it pays at least one of its members to provide some
amount of the public good unilaterally, that is, to bear the full cost of
providing it alone. Any group which is not privileged is said to be latent.
Where the group is privileged, there is, in Olson's view, a 'presumption'
that the public good will be provided; but there should be no such
presumption in the case of a latent group. Nevertheless, some latent
groups (in Olson's account, which is a little muddy at this point) are
sufficiently small that through some sort of strategic interaction amongst
their members they may succeed in providing some amount of the public
good. (They do not have so many members, says Olson, that an
individual contribution to the provision of the public good will go

introduction: the problem of collective action 9

unnoticed by other members.) Such groups are called intermediate. The
remaining latent groups are so large that this sort of strategic interaction,
depending as it does on individual contributions being 'noticeable', is
impossible and an individual will contribute only if there is a selective
incentive to do so, that is, the individual receives a (private) benefit if and
only if he contributes and/or incurs a (private) cost if and only if he fails
to contribute. Thus, for example, trade unions, which are founded
primarily to provide for their members certain public goods such as
higher wages and better working conditions, have also had to offer
prospective members sickness, unemployment and dispute benefits and
other positive selective incentives, and to operate a 'closed shop' which
bars non-members from employment.

In deciding whether or not to contribute or participate, the individual
compares the cost to him of making his contribution and the benefit to
him of the additional amount of the public good provided as a result of his
contribution. The final italicized phrase encompasses both the public
good which he himself directly produces or which is funded by his
contribution and whatever additional public good is provided by the
contributions that others may make as a result of his contribution
(because their contributions are in some way contingent on his). This
second component of his benefit may not be forthcoming, because the
required interdependence is absent. It is this interdependence which for
Olson characterizes 'intermediate' groups.

Now we can state the three arguments which Olson offers in support
of his argument that larger groups are less likely than smaller groups to
provide any (or an optimal) amount of the public good. 13

(i) The larger the group, the smaller is each individual's net benefit
from the public good.

(ii) The larger, the group, the less the likelihood that it will be privileged
or intermediate.

(iii) The larger the group, the greater the 'organization costs' of
providing the public good (including the costs of communication
and bargaining amongst group members and perhaps the costs of
creating and maintaining a formal organization).

The last of these claims is the most straightforward. It is also no doubt
empirically true, for very many cases.
The second claim is a little less straightforward. How much support it

10 introduction: the problem of collective action

gives to the argument that a public good is more likely to be provided in
smaller groups depends on the reliability of Olson's 'presumption' that
the public good will be provided in privileged groups and on how likely
it is that collective action will be successful in intermediate groups. Olson
says that the outcome of interaction in intermediate groups is 'in-
determinate'. As for the 'presumption', it is perhaps appropriate only
where there is just one individual who is willing to provide the public
good unilaterally (and even then there should be no presumption that an
optimal amount of it will be provided). But if two or more individuals are
so willing, then there could be strategic interaction amongst them . . .
and the outcome of such interaction is indeterminate. 14 (The game
amongst these players may be what is known as a Chicken game, which
will be discussed in some detail in the next chapter.) The privileged group
is therefore in.effect a group within which there is an intermediate group,
that is, a group with a subgroup whose members interact strategically.

The privileged group, it seems to me, is a special case of a group with at
least one subgroup whose members collectively find it worthwhile to
provide some amount of the public good by themselves, that is, a subgroup
such that, if all its members cooperated to provide the public good, each
of them would be better off than they would be if none of the public good
was provided. Again, this does not guarantee that any of the public good
will be provided, since normally there will be strategic interaction
amongst the members of the subgroup, and there will be strategic
behaviour of a different kind which may also obstruct provision of the
public good resulting from the coexistence of several such subgroups. A
group which is 'privileged' in this generalized sense is also, then, a group
within which there is at least one 'intermediate' subgroup.

A final point about claim (ii) is that, as Russell Hardin has observed,
there is no necessary connection, and probably a very weak correlation,
between the size of a group and whether it is privileged (in Olson's or my
generalized sense) or intermediate. Privileged groups can be large;
groups as small as two can be intermediate or latent. 15

It is worth emphasizing here parenthetically that it is dangerous to
distinguish intermediate and latent groups, as Olson sometimes did and
as so many later authors have done, by reference to whether an
individual contribution is 'noticeable' or 'perceptible'. Such talk has led a
number of writers astray. 16 Individual contributions can be perfectly
'noticeable' in a group which is not privileged and in which there is no

introduction: the problem of collective action 11

strategic interaction, and which therefore fails (in the absence of selective
incentives) to provide any of the public good; the failure arises because
each individual's contribution, though 'noticeable', brings too little of
the public good to be worth the cost of the contribution.

This leaves the first of Olson's three arguments about the effect of
increasing group size. As it stands this claim is undecidable. Before we
can assess it, we must know what kind of public good is involved and
what is held constant as size varies, for, as Hardin says, it is not possible
to increase size while holding everything else constant. 17 We should,
however, hold as many things as possible constant if we are to isolate a
pure size effect. Now the individual's net benefit can decrease as group
size increases because the costs of providing the public good (excluding
the organizational costs) increase or the individual's benefits decrease or
both. If it is a pure size effect we are looking for, we should count a cost
increase as support for Olson's claim only where such an increase is
unavoidable. The individual's benefit, on the other hand, decreases with
group size only if there is imperfect jointness or some degree o( rivalness
or both. If jointness is less than perfect, that is, there is some 'crowding',
then the amount available to an individual decreases as the number of
consumers increases. If there is rivalness, then, as size increases, the
individual's benefits decrease, whether or not the amount actually
available to him decreases. (Normally, rivalness is an effect of imperfect
jointness. But the two are analytically distinct, and in practice the effects
of rivalness can set in, as group size increases, before the effects of
imperfect jointness or literal 'crowding' do.) 18

Olson's first claim in support of the 'size' effect, then, is not necessarily
true. It holds only where costs unavoidably increase with size or where
there is imperfect jointness or rivalness or both. Most goods, however,
exhibit some divisibility, and most public goods interactions exhibit some
rivalness (which is, recall, a property of individual utility functions rather
than directly of the good). That is the theoretical position ; in practice, we
often want to compare groups which differ not only in size but in so
many other particulars that this claim is undecidable because isolating a
pure size effect is impossible.

There is a more important reason for not pursuing the issue here. The
argument here (following Olson and Hardin) assumes that we can
simply subtract costs from benefits. This is generally unrealistic (as
Olson himself admits 19 ). Preferences should instead be represented by

12 introduction: the problem of collective action

indifference maps. This will be done in chapter 2. Further, Olson's whole
analysis is entirely static: the individual is supposed in effect to make just
one choice, once and for all, of how much to contribute to the public
good. But in the real world, most public goods interactions are
dynamical. The choice of whether to contribute and how much to
contribute is a recurring one. There is interaction over time between
different individuals' choices. And the individual's intertemporal pre-
ferences (how much he discounts future relative to present benefits)
matter. A dynamical analysis is the subject of chapters 3 and 4.

Olson's model, then, is rather unrealistic. Accordingly, not too much
weight should be attached to conclusions derived from it, including
conclusions about the effects of increases in group size. The size effect
which I think should be taken most seriously is the increased difficulty of
conditional cooperation in larger groups. For, as we shall see, in a
dynamical analysis the provision of a public good, or collective action
more generally, requires that amongst at least some members of the
group there is conditional cooperation. Olson is of little help here, since
he does not provide (indeed cannot provide, within his static model) an
analysis of conditional cooperation or of any other sort of strategic
interaction over time.

To round out this brief discussion of Olson's treatment of the problem
of collective action, a comment is in order on his assumptions about
incentives and individual motivation. Recall that, according to Olson,
only a selective incentive will motivate the member of a large latent
group (one that is too large to be intermediate) to contribute to the
provision of the public good. (If Olson did not think of individual
contributions in such groups as being 'imperceptible' or 'infinitesimal',
he would perhaps have said : only the addition of a selective incentive will
make the difference between contribution and non-contribution. For
there is no reason why, upon the introduction of selective incentives, the
public good benefit to the individual should drop out of his calculation
even though it is very small : it is never so small as to be 'infinitesimal'. ) In
fact, says Olson, the public good lobbying efforts of large groups are by-
products of organizations which obtain their support by offering
selective incentives. 20 But this argument, as several writers have pointed
out, though it helps to explain the maintenance of the organization, does
not explain its origin.

Now selective incentives can be either positive or negative - providing

introduction: the problem of collective action 13

a benefit to a contributor or imposing a cost on a non-contributor - and
they are limited, in Olson's account, to either 'monetary' or 'economic'
incentives and 'social' incentives. The social incentives essentially derive
from the desire for approbation and the dislike of disapprobation, and
work through mechanisms like criticism and shaming by friends and
associates. Such incentives are effective only in relatively small groups.
Hence, a very large group might yet succeed in providing a public good if
it has a federal structure, for within the local branches or subgroups
social incentives can operate to maintain support. (And I would add : if
the local branches are small enough for social incentives to be effective,
they are probably small enough for conditional cooperation to be
sustained, perhaps with the help of the social incentives. More on this in
later chapters.)

There are therefore at most four components in the individual's
benefit-cost calculations: (i) the benefit to the individual from the
increased amount of the public good provided as a result of his
contribution; (ii) the cost of his contribution; (iii) the individual's
portion of the costs of organization; and (iv) the 'economic' and 'social'
benefits and/or costs which operate as selective incentives.

Olson explicitly excludes other types of incentives, including 'psycho-
logical' ones, such as 'the sense of guilt, or the destruction of self-esteem,
that occurs when a person feels he has forsaken his moral code'. 2 1 The
important reason why (in any explanatory theory) the range of
incentives which are assumed to motivate individuals must be limited -
though this is not among the reasons Olson gives for his restriction - is
that without such a limitation a rational choice theory such as Olson's is
liable to become tautologous. Three important kinds of motivation
which Olson - in common with nearly all other rational choice theorists
- excludes are altruistic motivations (to be discussed in chapter 5),
expressive motivations and 'intrinsic' motivation by benefits got in the
very act of participating in the provision of the public good as opposed to
the benefits which successful provision would bring. The last two give
rise to non-instrumental action. 22

The Prisoners' Dilemma

It has been widely asserted that individual preferences in public goods
interactions and in collective action problems generally are (or usually

14 introduction: the problem of collective action

are) those of a Prisoners' Dilemma game. 23 This game is defined as
follows.

Suppose that there are just two individuals (or players) and that each
of them may choose between two courses of action (or strategies). The
players are labelled 1 and 2 and the strategies C and D. The two players
must choose strategies simultaneously, or, equivalently, each player
must choose a strategy in ignorance of the other player's choice. A pair of
strategies, one for each player, is called a strategy vector. Associated with
each strategy vector is a payoff for each player. The payoffs can be
arranged in the form of a payoff matrix. The payoff matrix for the two-
person Prisoners' Dilemma which will be studied in this book is:

player 2

C D

x, x z, y

player 1

y, z w, w

where y > x > w > z. Throughout the book, the usual convention is
adopted that rows are chosen by player 1, columns by player 2, and that
the first entry in each cell of the matrix is the payoff to player 1 and the
second entry is 2's payoff.

Notice first that, since we have assumed y > x and w > z, each player
obtains a higher payoff if he chooses D than if he chooses C, no matter
what strategy the other player chooses. Thus, it is in each player's interest
to choose D, no matter what he expects the other player to do. D is said to
dominate C for each player.

However, notice now that, if each player chooses his dominant
strategy, the outcome of the game is that each player obtains a payoff w,
whereas there is another outcome (C, C), which yields a higher payoff to
both players, since we have assumed x > w.

Let us say that an outcome (Q) is Pareto-optimal if there is no other
outcome which is not less preferred than Q by any player and is strictly
preferred to Q by at least one player. An outcome which is not Pareto-
optimal is said to be Pareto-inferior. Thus, in the two-person Prisoners'
Dilemma, the outcome (D, D) is Pareto-inferior.

If the players could communicate and make agreements, they would
presumably both agree to choose strategy C. But this would not resolve

introduction: the problem of collective action 15

the 'dilemma', since neither has an incentive to keep the agreement:
whether or not he thinks the other player will keep his part of the
agreement, it pays him to defect from the agreement and choose D.

C and D are the conventional labels for the two strategies in the
Prisoners' Dilemma. They stand for Cooperate and Defect. I use them
throughout this book, though they are not entirely appropriate: one
player may 'Cooperate' (choose C) by himself, and he may 'Defect'
(choose D) even though no agreement has been made from which to
defect. In this book, Cooperation and Defection (with capital initials)
will always refer to strategies in a Prisoners' Dilemma (or, in chapter 2, in
some other game).

If communication between the players is impossible or prohibited, or
if communication may take place but agreements are not binding on the
players, then the game is said to be non-cooperative. The Prisoners'
Dilemma is defined to be a non-cooperative game. If it were not, there
would be no 'dilemma' : the players would obtain (C, C) as the outcome,
rather than the Pareto-inferior outcome (D, D). In the situations of
interest in this book, communication is generally possible but the players
are not constrained to keep any agreements that may be made. It is the
possibility of Cooperation (to achieve the outcome (C, C)) in the absence
of such constraint that will be of interest.

As a generalization of this two-person game, an iV-person Prisoners'
Dilemma can be defined as follows. Each of the N players has two
strategies, C and D, available to him. For each player, D dominates C,
that is, each player obtains a higher payoff if he chooses D than if he
chooses C, no matter what strategies the other players choose. However,
every player prefers the outcome (C, C, . . ., C) at which everybody
Cooperates to the outcome (D, D, . . ., D) at which everybody Defects.
Thus, as in the two-person game, every player has a dominant strategy
but if every player uses his dominant strategy the outcome is Pareto-
inferior.

Two-person and iV-person Prisoners' Dilemmas can both be defined
in the more general case when any finite number of strategies is available
to each player. The generalization, which could be made in several ways,
must at least have the characteristic that the predicted outcome is
Pareto-inferior. In particular, it could again be stipulated that every
player has a strategy which dominates each of the others, and if every
player uses his dominant strategy the outcome is Pareto-inferior. I shall

16 introduction: the problem of collective action

not elaborate on this here, as my discussion in this book will mainly be
confined to the two-strategy games, though in chapter 2 I shall also
consider games in which each individual can choose to contribute a
continuously variable amount within some range.

Let us go back now to the 'tragedy of the commons'. In Garrett
Hardin's account, each individual has in effect a dominant strategy: to
add an animal to his herd on the common, to discharge his sewage
untreated, to kill as many whales as possible and so on. Each of these
corresponds to strategy D. The alternative, strategy C, is to refrain from
doing these things. Hardin assumes, in effect, that D yields the highest
payoff to each individual, no matter what strategies the other individuals
choose (that is, no matter how many of them Cooperate); and he
assumes that every individual prefers the mutual Cooperation outcome
(C, C, . . ., C) to the mutual Defection outcome (D, £>,..., D). In other
words, individual preferences are assumed to be those of an iV-person
Prisoners' Dilemma.

Russell Hardin has argued explicitly that public goods interaction in
sufficiently large groups in fact 'the collective action problem' more
generally - can be represented by the JV-person Prisoners' Dilemma. 24
His analysis is as follows. Suppose that each of N individuals has the
choice (and only the choice) between contributing and not contributing
one unit of the cost of producing a non-excludable good (one unit of a
numeraire private good) and that every unit contributed produces an
amount of the public good with benefit r. Suppose that each individual's
utility is nr/N if he does not contribute and nr/N — 1 if he does, where n is
the total number of units contributed. (Notice that this means that the
public good exhibits some rivalness: each individual's utility declines
with increasing N, which is the number of individuals who actually
consume the good, since nobody is excluded.) Then, if m other
individuals contribute, an individual's utility is mr/N if he does not
contribute and (m + l )r/N — 1 if he does. Thus, the first of these utilities
exceeds the second if and only if N > r, which is independent of m. In
other words, no matter how many other individuals contribute, it does
not pay anyone to contribute as long as the size of the public (N) exceeds
the ratio of benefits to costs (r). When N > r, the game is an N-person |
Prisoners' Dilemma (as defined above): each individual has a dominant ,
strategy, and the outcome which results when everyone chooses his
dominant strategy is for everyone less preferable than another outcome.

introduction: the problem of collective action 17

But when N < r, the dominant strategy for every individual is to
contribute and the resulting outcome is the only Pareto-optimal
position.

This argument, if correct, would apply also to the public goods with
which Hobbes was chiefly concerned, namely domestic peace and
security and national defence. I shall indeed show (in chapter 6) that
Hobbes assumed men's preferences in the absence of the state to be those
of a Prisoners' Dilemma game. The remainder of Hobbes's theory can
then be summarized, somewhat crudely, as follows : (a) in the absence of
any coercion, it is in each individual's interest to choose strategy D; the
outcome of the game is therefore mutual Defection; but every individual
prefers the mutual Cooperation outcome ; (b) the only way to ensure that
the preferred outcome is obtained is to establish a government with
sufficient power to ensure that it is in every man's interest to choose C.

This is the argument which I wish to criticize in this book. But there is
one element of the argument which I shall not quarrel with, namely, the
analysis of the Prisoners' Dilemma given above. If individual preferences
in the provision of a public good are in fact those of a Prisoners'
Dilemma, then it is quite correct to conclude that the players will not
voluntarily Cooperate. To avoid any misunderstanding, I emphasize
that the conclusion is correct no matter what the entries in the payoff
matrix (which is assumed to be a Prisoners' Dilemma) actually
represent, just as long as it is assumed that each player is concerned only
to maximize his own payoff. Of course, the payoffs may not reflect all the
incentives affecting the individuals in the situation in question. The
conclusion still follows logically; but it is possible to argue that the
payoff matrix is a poor description of the relevant real world situation
and that in reality the players do Cooperate, because the omitted
incentives are more important than those reflected in the payoff matrix.

In the next three chapters, the payoffs are assumed not to reflect, inter
alia, (i) incentives due to external coercion, including that applied or
threatened by the state or any other external agency or by other
members of the group (apart from the tacit threats and offers which may
be thought to be embedded in conditional Cooperation - about which
more later); (ii) altruistic motivation; and (iii) any 'internal sanctions'
like guilt, loss of self-respect and so on, which may result from failure to
conform to a norm, live up to one's own ideals, perform one's duties, or
whatever. In chapter 5 I shall begin with a matrix of payoffs which again

18 introduction: the problem of collective action

do not reflect these three classes of incentives, but then I shall consider
the effects of assuming that individuals are altruistic (that is, they take
account of other players' payoffs as well as their own in choosing
strategies).

The expression 'voluntary Cooperation', used occasionally through-
out the book, refers to Cooperation chosen only on the basis of the
matrix of payoffs (or utilities, where the individual is in some way
altruistic); thus, voluntary Cooperation is Cooperation which, amongst
other things, is not the result of external coercion, including that applied
or threatened by the state.

The problem of collective action

I shall argue in the next chapter that, in many interesting problems of
public goods provision, individual preferences at any point in time are
not those of a Prisoners' Dilemma. Many other preference structures can
arise. These include Chicken and Assurance games, whose two-person
payoff matrices are shown below.

3, 3

2,4

4,4

1,2

4,2

1, 1

2, 1

3, 3

Chicken Assurance

Surely, then, we should not equate 'the problem of collective action' with
the Prisoners' Dilemma, as many writers have done - even though some
of these alternative representations of public goods interaction (most
notably the Assurance game) do not seem to present 'problems' in the
sense which I think most people intend by use of the expression
'collective action problems'. What then do we mean by this expression?

Jon Elster gives a 'strong definition' of the collective action problem,
which identifies it with the Prisoners' Dilemma, and a 'weak definition'
which requires that (i) universal cooperation is preferred to universal
non-cooperation by every individual (as in the Prisoners' Dilemma) and
(ii) cooperation is 'individually unstable' and 'individually inacces-
sible'. 25 There is individual instability if each individual has an
incentive to defect from universal cooperation, and there is individual
inaccessibility if no individual has an incentive to move unilaterally from

introduction: the problem of collective action 19

universal non-cooperation. But then he points out that there are cases in
which cooperation is either individually unstable or individually
inaccessible but not both - for example Chicken and Assurance games -
but which nevertheless present collective action problems (though 'less
serious' ones in the case of Assurance games).

The definition which I think gathers up all the cases that Elster and
others are actually concerned with is that a collective action problem
exists where rational individual action can lead to a strictly Pareto-
inferior outcome, that is, an outcome which is strictly less preferred by
every individual than at least one other outcome. The problem with this
definition - an unavoidable problem, it seems to me, if one wants to give
a general definition that covers all the cases one intuitively thinks of as
collective action problems - is that it's not clear in some situations what
rationality prescribes (even if we rule out, as I am assuming we should do
here, notions of rationality not considered by game theorists). This is
true of Chicken games. Any outcome of a Chicken game, including the
Pareto-inferior mutual Defection outcome, can be rationalized. Hence,
rational action can plausibly lead to a Pareto-inferior outcome, so that
on my account it is a collective action problem.

Whether Assurance games are collective action problems again
depends on what one takes rationality to prescribe. I shall take the view
that, if a game has multiple equilibria (as the Assurance game does) but
one of them is strictly preferred to all the others by everyone, then the
Pareto-preferred one will be the outcome. On this view, rational action
in an Assurance game does not lead to a Pareto-inferior outcome, so that
this game is not a collective action problem.

Since preferences in some public goods interactions are those of an
Assurance game, not all such interactions are collective action problems.

In the case of the (one-shot) Prisoners' Dilemma, rational action
unequivocally leads to a Pareto-inferior outcome, so on my account all
situations representable as Prisoners' Dilemmas are collective action
problems. So are many other games (some of which will be encountered
in the next chapter). Of course, not all of these games (including the
Prisoners' Dilemma and Chicken games) correspond to public goods
interactions.

Elster has said that politics is 'the study of ways of transcending the
Prisoners' Dilemma'. 26 In the light of this discussion of 'the collective
action problem' (and in anticipation of the discussion of alternatives to

20 introduction: the problem of collective action

the Prisoners' Dilemma in the next chapter), I think we should be a little
more expansive and say that politics is the study of ways of solving
collective action problems.

Time and the lone exploiter

It's worth noting parenthetically that the degradation of a 'common'
may not be the result of failure to solve a collective action problem. It
may occur even where the common has only one user and he acts
rationally.

The 'tragedy of the commons', on Garrett Hardin's account, arises
because, at any point in time, each individual finds it in his interest to
exploit the common (choose strategy D) no matter what the others do.
The 'tragedy' does not arise, as some people have written, because each
man reasons that 'since the others are going to ruin the common anyway,
I may as well exploit it too'. (In fact, if the others do not exploit the
common, if they restrain themselves and choose strategy C, then each
individual will find it even more profitable to exploit it than if they do.) It
cannot be said, then, that the common would not be ruined if only one
individual had access to it ; that if a lake and all its lakeside factories were
owned by one man, he would treat his wastes before discharging them
into the lake; that if one man had an exclusive right to kill whales, he
would see that they did not become extinct.

But surely, it may be said, the sole hunter of whales would not kill
them all off, for his whole future livelihood, or at least all his future
profits, depends on their survival. Unfortunately, this may not be the
case.

Consider a common which one man has exclusive rights to exploit
without restraint, and suppose now that at some point in time he is
contemplating his whole future course of action with respect to this
common. Let us suppose that he divides the future into equal time
periods (months, years or whatever) and that in each time period he will
receive a payoff. The sequence of payoffs he will receive depends on the
course of action he chooses (for example, how many whales he kills in
each period). Clearly, what he chooses to do will depend on the present
value to him of future payoffs. At one extreme he may place no value
whatever on any payoff except the one in the time period immediately
before him. In this case, the prospect of zero payoffs from some point in

introduction: the problem of collective action

the future onwards (as a result of the extinction of the whales, for
example) does not trouble him at all. He will act in each time period so as
to maximize his payoff in the current time period, and the result may be
the ruin of the common.

It is generally assumed that future payoffs are exponentially dis-
counted to obtain their present values. In the case when future time is
divided into discrete periods, this means that the present value of a
payoff X, to be made t time periods from the present is X,a\ where a is
a number such that < a < 1 and 1 - a is called the discount rate. The
higher the discount rate, the lower the present value of future payoffs. If,
for example, the individual's discount rate is 0.1 (that is, a =0.9), then a
payoff worth 100 units if received now would have a present value of 90 if
it were to be received one period hence, 81 if it were to be received two
periods hence and so on.

Intuitively, we should expect that if the discount rate is sufficiently
high, then an exploiter who is seeking to maximize present value may
eventually and quite 'rationally' ruin the common, even in the absence of
other exploiters. The simple mathematics of this are set out by Colin
Clark in his study of the exploitation of renewable resources (which, it
should be remembered, include atmospheric, soil and water resources as
well as such things as whales, fish and bison). 27 In the case when the
resource of the common is a biological population, the discount rate
which is sufficiently high to result in the extinction of the population will
depend above all on the reproductive capacity of the population. (In
Clark's model, this is all it depends on.)

The ruin of the common by a single individual, though it may be
unfortunate, is not a 'tragedy' in Hardin's sense. (In the 'tragedy of the
commons', the tragedy resides in the fact that 'rational' action on the part
of each individual brings about a state of affairs which nobody wants.)
Nor would it be a 'tragedy' if several individuals with similar preferences,
including a shared high discount rate, ruined the common together, for
this outcome would not be Pareto-inferior for them.

Solutions? Community, states, entrepreneurs, property
rights and norms

There are, broadly speaking, two sorts of solution to collective action
problems, which I will call 'spontaneous' or 'internal' solutions and

22 introduction: the problem of collective action

'external' solutions. Internal solutions neither involve nor presuppose
changes in the 'game', that is, in the possibilities open to the individuals
(which are in part determined by the 'transformation function', specify-
ing how much of the public good can be produced with a given
contribution), the individuals' preferences (or more generally attitudes),
and their beliefs (including expectations). External solutions, on the
other hand, work by changing the game, that is, changing people's
possibilities, attitudes or beliefs. The changes do not necessarily
originate outside the group of individuals who have the collective action
problem. Since individual action is the product directly of the
individual's possibilities, attitudes and beliefs, these two exhaust the
possible sorts of solution.

It could be said that in the case where an internal 'solution' is
forthcoming, there was no 'problem' there to solve. For example, if the
'problem' is correctly modelled as a dynamic game which, though it
consists let us say of an iterated Prisoners' Dilemma, is not itself a
Prisoners' Dilemma and as a consequence the outcome produced by
rational egoists (without any external assistance or other interference)
would be mutual cooperation throughout the game, then it could be said
that preferences (including intertemporal preferences), etc., are such that
there is no collective action problem. This would be a perfectly
reasonable use of the word problem, but I shall not adopt it here. In fact,
I shall take the view that the internal solution is the basic one, in two
connected senses. It is, first, the only one which is complete in itself. All
the external solutions presuppose the prior and/or concurrent solution
of other problems, usually (always?) of collective action problems. Many
of them, for example, involve the use of threats and offers of sanctions,
and the creation and maintenance of the sanction system entail the prior
or concurrent solution of collective action problems. (Why, for example,
should the rational egoist pay his portion of the taxes that the state
requires to maintain its police forces, etc., or why should the individual
member of a community go to the trouble of punishing a free rider when
he could be a free rider on the sanctioning efforts of others?) The internal
solution is basic in a second sense : until we know whether a solution of
this kind is possible and what form it will take, we cannot say what work,
if any, remains to be done by other putative solutions. Thus, understand-
ing the prospects for and obstacles in the way of an internal solution
helps us to see what sorts of external solution are necessary and are likely
to emerge in a given context.

introduction: the problem of collective action 23

External solutions can themselves be divided into two broad cate-
gories, which for short I will call centralized and decentralized ; or, better,
they can be arrayed along a continuum running from perfectly
centralized to perfectly decentralized. Combinations of them are
possible - normal, in fact. A solution is decentralized to the extent that
the initiative for the changes in possibilities, attitudes or beliefs that
constitute an external solution is dispersed amongst the members of the
group; or, the greater the proportion of the group's members involved in
solving the collective action problem (e.g. applying sanctions to free
riders), the more decentralized the solution. Contrariwise, a solution is
centralized to the extent that such involvement is concentrated in the
hands of only a few members of the group.

Centralized solutions are typified, of course, by the state, while
decentralized solutions characterize community. I have devoted another
book to the ways in which a community can provide itself with public
goods without the help of the state and will not reproduce the arguments
here. 28 By a community I mean a group of people (i) who have beliefs
and values in common, (ii) whose relations are direct and many-sided
and (iii) who practise generalized as well as merely balanced reciprocity.
The members of such a group of people, or all of its active adult
members, can wield with great effectiveness a range of positive and
negative sanctions, including the sanctions of approval and disapproval
- the latter especially via gossip, ridicule and shaming. Decentralized
solutions can sometimes be effective where there is little community, but
the size of the group would still have to be relatively small (as it must be
in a community).

External solutions are not necessarily restricted to the use of threats
and offers of positive and negative sanctions. These, it is true, work not
by altering an individual's preferences among outcomes (properly
defined) but by altering his expectations about the actions to be taken by
others (and hence the expected utility associated with alternative courses
of action). But there are other ways in which an individual's expectations
about others' behaviour can be altered and other ways in which he can
be got to contribute to a public good, without the use of threats and
offers or of force, whether centralized or decentralized. These include
persuasion - providing information and arguments about the alternat-
ives, about the consequences of adopting the various courses of action,
about others' attitudes and beliefs and so on. Such methods are
characteristic of the political entrepreneur, an external solution (relat-

24 introduction: the problem of collective action

ively centralized, though usually closely combined with decentralized
mechanisms) which I shall discuss shortly.

My main concern in this book is with the internal solution - with the
possibility of spontaneous cooperation - as an alternative to the state.
But before turning to this, I want to comment briefly, first, on the role of
the political entrepreneur in the solution of collective action problems,
and secondly, on the claims made by a number of writers that certain
collective action problems can be solved by establishing private property
rights and by norms.

Political entrepreneurs

In what sense do political entrepreneurs or leaders 'solve' collective
action problems? In general, to solve or remove a collective action
problem he or she must of course change individual preferences (or more
generally attitudes), or change beliefs (including expectations) or inject
resources (very probably knowledge, or new technology, like guns) into
the group so as to make its members' efforts more productive.

Merely offering his services (working to obtain the public good) in
exchange for support (subscriptions, food and shelter, or whatever) does
not in itself constitute a distinctive solution to the problem. For, in the
first place, the entrepreneur's services are themselves a public good, so
that supporting him also gives rise to a collective action problem. This
includes the case of the politician who in seeking electoral support offers
his constituents legislative or other changes they favour. The collective
action problem his potential supporters had in obtaining the public
goods which such changes would have brought them is replaced by the
collective action problem of getting him elected. And secondly, if the
entrepreneur gains support by offering selective incentives, as well as by
promising to work for the public good, then the solution is precisely the
one proposed by Olson himself, in his 'by-product' theory. 29

In many interesting cases the political entrepreneur may require little
or no support from the members of the group whose collective action
problem is at issue, because he is supported by (i.e., brings resources
from) some external source. He might, for example, in his efforts to solve
a local collective action problem, be supported by a pre-existing
organization (the Communist Party, say, or the Catholic Church). This
makes it easier to explain why the local problem is solved (for the
members of the local group do not have to produce a 'surplus' to pay or

introduction: the problem of collective action 25

feed the entrepreneur), but it leaves unexplained (a) the production of the
resources brought in by the political entrepreneur, which will usually
entail that a prior collective action problem - for example the creation
and maintenance of an organization - has been solved; and (b) how,
even though the (local) group does not have to support the entrepreneur,
it now manages to solve a collective action problem that it could not
solve without him. If the only difference the entrepreneur makes is the
addition of selective incentives to their benefits, then, once again, we do
not have a distinctive solution.

But the political entrepreneur is not just 'an innovator with selective
incentives', 30 or someone who simply concentrates or centralizes
resources. What is perhaps more characteristic of political entrepreneur-
ship is its role in changing beliefs - beliefs about the public good itself,
about what others have done and are likely to do and about others'
beliefs. Above all, we must remember that most collective action must
involve some form of conditional cooperation, for at a minimum an
individual would not cooperate if nobody else did. And as we shall see (in
chapters 3 and 4) conditional cooperation is a very precarious business.
It requires amongst other things that the conditional cooperators have
information about others' behaviour. The required monitoring can be
done by the political entrepreneur.

The entrepreneur can also try to persuade people that their contri-
butions make a big enough difference, either directly or indirectly
through their effect on others' behaviour. The second of these might be
achieved by persuading people that others' efforts are contingent on
theirs.

An organization whose aim is to provide public goods for a very large
group might be able to expand its membership and achieve its aims by
having its cadres work to solve, through any or all of these en-
trepreneurial methods, smaller-scale collective action problems for
much smaller subgroups. A nationwide movement, for example, may be
built upon the success of its cadres in solving local collective action
problems and bringing tangible benefits quickly. Samuel Popkin has
given an excellent account of activities of this sort in Vietnam, showing
how four politico-religious movements (the Catholic Church, the Cao
Dai, the Hoa Hao and the Communist Party) won support by having
their cadres help the villages, both by providing selective incentives and
by facilitating cooperation in the provision of public goods. 31 These

26 introduction: the problem of collective action

private and public goods - with varying degrees of indivisibility and
excludability - included the provision of educational opportunities; the
creation of insurance and welfare systems; agricultural improvements;
the establishment of stock-farm cooperatives; improvements in water
storage and irrigation facilities; the creation of local courts to arbitrate
disputes; and protection against French courts, marauding notables and
local landlords.

Property rights

Many economists, and nearly all those of the 'property rights' school,
believe that the solution to free rider problems in public goods provision,
and in particular those which would lead to the over-exploitation of a
'common property resource', lies in the establishment of private
property rights. Without such rights, the argument goes, every in-
dividual has an incentive to intensify his use of the resource because (as
we saw in discussing Garrett Hardin's 'tragedy of the commons')
although, with each increment in use, every unit of his (and everybody
else's) input becomes slightly less productive, this is up to a point
outweighed by the marginal return from the increased input. Intensify-
ing use of the resource is continued up to the point where all the 'rent'
(income or other return) from the resource has been dissipated.
Likewise, the benefits arising from any improvement or renewal or other
investment he might make in the resource would be shared by all the
users while the costs would be borne by the individual alone. There will
therefore be overuse and underinvestment. With the establishment of
private property rights, however, the external effects of each individual's
actions are 'internalized' : all the costs of an increase in use of the resource
are borne by the individual, as are all the benefits of investing in its
conservation or improvement.

The argument that the 'tragedy of the commons' is the fate of common
property resources, and that overuse or underinvestment will be avoided
only if common property rights are displaced by private property rights,
seems to be positively mocked by the facts. The commons of the
European open field system, far from being tragically degraded, were
generally maintained in good health during the whole of their lifetimes of
many hundreds of years. There is a detailed study of a Swiss alpine
village (not, of course, operating an open field system) whose members
have for more than five hundred years possessed and used in common

introduction: the problem of collective action 27

various resources, including mountain-side pastures, side by side with
privately owned land and other resources and during all this time the
productivity of the common land has been maintained and much effort
has been invested in its improvement. 32 Contrast with this the treatment,
especially in recent decades, of much privately owned land by its very
owners: the destruction of vast tracts of rain forest for the sake of a few
profitable years of ranching; or the set of practices which together are
causing the loss of topsoil from cultivated land through wind and water
erosion on such a scale that, according to a recent report, there will be a
third less topsoil per person by the end of the century. 33 In parts of
Africa, and elsewhere in the world, overexploitation of grazing lands has
been caused not by common property arrangements per se but by their
destruction or disruption. 34 There are, as we saw earlier, perfectly good
reasons why the rational private owner or user of a resource might
knowingly destroy it; in particular, he might place a very low value on
benefits to be derived from the resource in the distant as opposed to the
immediate future.

Where do the property rights economists go wrong? 35 In the first
place, many of them do not distinguish common property in a resource
from open access to it. 'Communal rights', say Alchian and Demsetz, '. . .
means that the use of a scarce resource is determined on a first-come,
first-serve basis and persists for as long as a person continues to use the
resource'. 36 This is wrong, or at least an abuse of language. If there is
open access, then nobody is excluded from using the resource and there
is no regulation of the activities of those who do use it. But if there is
common ownership or collective control of the resource, then the
members of the collectivity, whatever it is, can regulate its use. This is
what happened in the European open field system, where the villagers
rigorously excluded outsiders from use of the various commons they
owned or possessed collectively, and carefully regulated insiders' use,
typically by allotting to individuals 'stints' in proportion to their
(privately owned) arable holdings and punishing people for infringe-
ments. The alpine community described by Netting similarly practised
strict external and internal regulation of its commons. So too have
countless 'primitive' collectivities and peasant villages all over the world.

It is to resources with open access, not to 'common property
resources', that the property rights economists' argument about over-
exploitation and underinvestment applies. It is not a matter of establish-

28 introduction: the problem of collective action

ing the right sort of property rights, of moving from collective to private
property rights. It is rather a matter (at this stage of the argument at
least) of establishing property rights where there were none; for property
entails exclusion, so that where there is open access to a resource, there is
no property in it. 37

The property rights economists tend to see only two or three
possibilities: open access and private property, to which is sometimes
added state ownership. But almost any group of individuals can own or
possess property collectively. Historical and contemporary examples
are: a family; a wider kin group, such as a matrilineage; all those in a
village who also possess land privately; a band; an ethnic group. Where
the property rights economists do notice common property rights, they
then argue that the costs of negotiating agreements regulating use and, if
agreements are forthcoming, the costs of policing them, will be very
great, and in this respect, common property rights compare un-
favourably with private property rights. 38 But there is no necessary
reason why transaction costs of all kinds should in total be greater in the
case of common property rights than in the case of private property
rights - and in the case of the open field system it was in fact the other
way round, essentially because of economies of scale in pastoral
production. 39

Finally, the property rights economists, having generally failed to
notice common property (as opposed to open access) and to study how
individual rights in it are guaranteed, tend to assume that property rights
must be enforced by the state. 40 But there can also be decentralized
enforcement or maintenance of property rights - both private and
common. (The sense of 'decentralized' intended here is the same as that
used in the general remarks made earlier on the solution of collective
action problems.) If a collectivity itself is to enforce its members' private
property rights or their rights to use the common property, then it must
of course be able to wield effective sanctions - unless the property rights
are respected as a result of 'spontaneous' conditional cooperation. If the
collectivity is a community, then, as we have seen, conditions are
conducive to conditional cooperation, and if this fails the community's
members have at their disposal a range of effective sanctions. The joint
owners of the commons in European open field villages, for example,
were communities in the required sense.

Enough has now been said, I think, to show that, insofar as the

introduction: the problem of collective action 29

solution of collective action problems is concerned, nothing new is added
by the introduction of property rights per se. An individual has property
in something only if others forbear from using it, and the forbearance is
the result of the threat or offer of sanctions, centralized or decentralized
(or of conditional cooperation - unless this be reckoned also to involve
threats and offers). It is the threats and offers of sanctions (and/or
conditional cooperation) that is solving the collective action problem, if
it is solved at all. Furthermore, as I remarked in an earlier section, the use
of some of these sanctions presupposes the solution to prior collective
action problems (for example, the formation and maintenance of a
state!).

Norms

There is, finally the suggestion that norms solve collective action
problems. I will comment on this very briefly, for my reaction to it is
similar to my view of the suggestion that the introduction of private
property rights solves collective action problems, and both follow from
the general remarks about solutions to these problems made in an earlier
section (though I shall not argue, as some have done, that property rights
are norms). The view that norms solve collective action problems - or
more precisely that they solve, amongst other things, the problems
inherent in 'generalized PD-structured situations' and coordination
problems - has been expounded by Edna Ullman-Margalit. 41 I shall
take it that a norm is generally conformed to and is such that non-
conformity, when observed, is generally punished. It is unclear whether
this is what Ullman-Margalit means by a norm, but in any case it is fairly
clear from her discussion of 'PD norms' that it is only 'a norm, backed by
sanctions' or 'a norm . . . supported by sufficiently severe sanctions' that is
capable of solving Prisoners' Dilemma problems. 42 So norms alone -
mere prescriptions for action that people generally conform to - do not
solve these problems.

If a norm is generally observed simply because it pays the individual to
do so (in the absence of sanctions), then there is no (collective action or
other) 'problem' to be solved in the first place. This would be the case if
the norm had been 'internalized'. I take this expression to indicate that
conformity to the norm does not require the application of external
sanctions, inducements or any other considerations; as a result of the
norm being internalized, the individual prefers to conform (without the

30 introduction: the problem of collective action

threat of punishment) or at least has some sort of motivational
disposition to do so. But then, as I say, we would not say that there was a
Prisoners' Dilemma or collective action 'problem' to be solved: the
individual preferences would not be those of a Prisoners' Dilemma or
would not be such as to lead to a collective action problem. Of course, we
might nevertheless wish to explain how the norm came to be internalized
or how people came to have such preferences.

If, on the other hand, a norm is generally observed because non-
conformity, when noticed, is generally punished, then it is the sanctions
that are doing the real work of solving the Prisoners' Dilemma or
collective action problem. The sanction system can of course be
centralized or decentralized, in the way discussed in an earlier section.
And again, it remains to be explained how the system of sanctions itself
came into being and is maintained. To this problem, the general point
made earlier about sanction systems applies: the maintenance of a
system of sanctions itself constitutes or presupposes the solution of
another collective action problem. Punishing someone who does not
conform to a norm - punishing someone for being a free rider on the
efforts of others to provide a public good, for example - is itself a public
good for the group in question, and everyone would prefer others to do
this unpleasant job. Thus, the 'solution' of collective action problems by
norms presupposes the prior or concurrent solution of another collective
action problem. And as my earlier remarks make clear, this would still be
the case if the sanctions were wielded by the state or by a political
entrepreneur.

Plan of the rest of the book

My purpose in this book is to examine the possibility of voluntary
cooperation in the provision of public goods and in the solution of other
collective action problems, and in doing so - and in other ways to raise
questions about what I take to be the most persuasive justification of the
state. The detailed study of voluntary cooperation which follows
(chapters 2, 3 and 4) can be read - and evaluated - independently of the
critique of the liberal theory of the state. Both as a study of cooperation
and as a study of the state and its alternatives, it is obviously far from
complete ; another part of the story is tackled in my Community, Anarchy
and Liberty, which is complementary to this book.

introduction: the problem of collective action 31

As a critique of the liberal justification of the state, the argument will
be in three stages. First, I argue in chapter 2 that in public goods
interactions the individual preferences at any point in time are not
necessarily those of a Prisoners' Dilemma game. This is true, I shall
argue, of both two-person and JV-person games and of cases where
strategy sets are continuous as well as those where individuals have only
two strategies available to them. It will emerge that important classes of
public goods provision problems are better represented by Assurance
and especially Chicken games, and in the continuous case by hybrids of
these two. In all these games, arguably, if the game is played only once,
some cooperation is more likely to be forthcoming than in cases for
which the Prisoners' Dilemma is the appropriate model.

In the next two chapters (3 and 4), however, I shall assume the worst:
that preferences at any point in time are those of a Prisoners' Dilemma
game. But I then go on to show that if time is introduced and the problem
is treated more dynamically, under certain circumstances voluntary
Cooperation is rational for each player, even assuming that he seeks to
maximize only his own payoff.

My argument here will be cased in terms of the Prisoners' Dilemma
supergame. This is the game consisting of an indefinite number of
iterations of one of the Prisoners' Dilemma games (two-person and N-
person) which were denned earlier. In each constituent game (as the
repeated game is now called), players choose strategies simultaneously,
as before, but they know the strategies chosen by all other players in
previous games. Each player discounts future payoffs; his discount rate
does not change with time, but discount rates may differ between players.
The constituent game is assumed not to change with time. (It would be
desirable to relax this last assumption in a more general treatment, and
permit the payoff matrix to change with time. See the final section of
chapter 4 below.)

The really important difference between the one-shot game and the
supergame is that players' strategies can be made interdependent in the
latter but not, of course, in the former, since players must choose
strategies simultaneously or in ignorance of each other's choices. In the
supergame, a player can, for example, decide to Cooperate in each
constituent game if and only if the other player(s) Cooperated in the
previous constituent game. It is on this possibility, the possibility of
using conditional strategies, that the voluntary Cooperation of all the
players turns.

32 introduction: the problem of collective action

Finally, in chapter 7, 1 shall raise doubts about the way in which this
justification of the state is approached. It is an essential and fundamental
feature of the theory I am criticizing that it takes individual preferences
as given and fixed. In particular, it is assumed that the state itself has no
effect on these preferences. This rules out ab initio the possibility,
amongst many others, that the state may exacerbate an already existing
collective action problem or create such a problem where none existed
before: that the state may affect, in other words, the very conditions
which are supposed to make it necessary. If preferences may change,
especially as a result of the activities of the state itself, it is not at all clear
what is meant by the desirability of the state.

Criticisms of this sort can of course be levelled against any theory
which is founded on assumptions about fixed individual preferences (as
most of economic theory and some polotical theory is); but they are
especially important, it seems to me, when the theory purports to justify
an institution (like the state) and when the theory is to apply to a very
long period of time (as a theory used to justify the state or to explain its
origin must do).

I have said that the arguments which are the object of my criticisms in
this book have been set out most explicitly by Thomas Hobbes. I shall
therefore give (in chapter 6) an exposition of his political theory. My
chief reason for devoting to this exposition a rather long chapter later in
the book, rather than a short summary at the start of the book where it
would otherwise belong, and for making what would otherwise be an
unpardonable addition to the considerable critical literature on Hobbes,
is that I think it is illuminating to look at these theories in terms of some
of the ideas presented in the earlier chapters on the Prisoners' Dilemma
and its supergame. I have asserted rather baldly in this informal
Introduction that Hobbes's theory is about non-Cooperation in Priso-
ners' Dilemma games (other writers have made similar assertions,
equating Hobbes's theory with, for example, Hardin's analysis of the
'tragedy of the commons'); but the story is more complicated and more
interesting than this and deserves a fuller account.

I shall also consider, more briefly, David Hume's political theory. For
although it is very similar to Hobbes's theory (despite Hume's objections
to what he took to be a fundamental element of Hobbes's theory, the idea
of the social contract) and although it is generally less rigorous than
Hobbes's version (in Leviathan), it does partly supply a deficiency in

introduction: the problem of collective action 33

Hobbes's treatment, namely that it is too static. Hobbes in effect treats
only a one-shot Prisoners' Dilemma game, whereas Hume's treatment is
more dynamic, with the discounting of future benefits playing an
important role. Also, in Hume, but not in Hobbes, there is explicit
recognition of the effects of size, a partial anticipation of Olson's 'logic of
collective action'.

Some of the ideas I am interested in here appeared much earlier than
Leviathan (above all in the Book of Lord Shang and the works of Han Fei
Tzu which were written in China in the fourth and third centuries bc),
but it was Hobbes and Hume who gave the first full, explicit statements
of the argument. And in later political theorists the argument is not
always explicit, does not stand out boldly and is less precise and less
coherent.

In Leviathan, Hobbes seems to assume that each man seeks to
maximize not merely his own payoff, but also his 'eminence', the
difference between his own and other people's payoffs. Hume, on the
other hand, assumes that most people are chiefly concerned with their
own payoffs but are also possessed of a limited amount of 'benevolence'.
In both cases, individuals take some account of other individuals'
payoffs; I call this 'altruism'. The effects of various sorts of altruism on
the outcomes of Prisoners' Dilemma games are treated briefly in chapter
5. Some of the material in that chapter will be of use in the discussion of
Hobbes and Hume and also in the final chapter.

2. The Prisoners' Dilemma, Chicken and
other games in the provision of public
goods 1

It has often been said - and indeed is still being said - that 'the problem of
collective action and the Prisoner's Dilemma are essentially the same' 2
(at least where the group is large or where it is not privileged) and that the
Prisoners' Dilemma game is the appropriate model of public goods
provision. Neither of these is the case. The argument against the first
claim - which has been made by a number of writers, most explicitly
perhaps by Russell Hardin 3 - was begun in the last chapter. Against the
second claim I argue in this chapter that the Prisoners' Dilemma is not
the only applicable game in the study of public goods provision. Since
many (though not all) important collective action problems arise in
connection with the provision of public goods, this chapter, in attacking
the second claim, will also attack the first.

More specifically, I shall argue that in public goods interaction the
individuals' preferences at any point in time are not necessarily those of a
Prisoners' Dilemma. I shall argue this, first, in the case where the
individuals choose between just two strategies, Cooperate (or contribute
to the provision of the public good) and Defect (not contribute), and
then in the case where each individual can choose to contribute a
continuously variable amount within some range (or choose from a large
number of discrete amounts which may be approximated by a continu-
ous variable). We shall see in particular that important classes of public
goods provision problems are better represented by the game of
Chicken, both in the two-strategy, two-person case and in the two-
strategy, AT-person case to which Chicken can be generalized, and that
structures of preferences can also be 'Chicken-like' in the case when the
strategy sets are continuous.

I emphasize that this chapter does not begin to consider the dynamics
of choice in public goods interaction ; it is concerned only with individual

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 35

preferences at one point in time and with choices in one-shot games. 4
Genuinely dynamical considerations enter into the analysis in the next
chapter, which considers repeated plays of the Prisoners' Dilemma, or
'supergames'. These supergames are themselves typically not Prisoners'
Dilemmas. So this chapter and the next two will establish (amongst
other things) that preferences in public goods interaction at a point in
time are often not those of a Prisoners' Dilemma and that, even if they
are, the 'dynamic' or intertemporal preferences of the resulting super-
game are usually not.

Alternatives to the Prisoners' Dilemma

If a 2 x 2 game is to be a Prisoners' Dilemma, then amongst other things
each player must (a) prefer non-Cooperation if the other player does not
Cooperate, and (b) prefer non-Cooperation if the other player does
Cooperate. In other words: (a') neither individual finds it profitable to
provide any of the public good by himself; and (b') the value to a player
of the amount of the public good provided by the other player alone (i.e.,
the value of being a free rider) exceeds the value to him of the total
amount of the public good provided by joint Cooperation less his costs
of Cooperation. Of course a player could not even form these preferences
if it were not possible for each player to provide some of the public good
alone. For many important public goods, I shall argue, either or both of
the conditions (a') and (b') fail, and sometimes even this precondition for
forming the preferences in question may fail.

If (a') fails for at least one of the players - if one of them has an
incentive to provide some amount of the public good even if he alone has
to pay the full costs - then we have what Olson calls a 'privileged' group.
In this case there is, according to Olson, a 'presumption' that the public
good will be provided by the players. If only one of the players is willing
to act unilaterally in this way, this presumption is reasonable. But if both

3, 3

1,4

4, 1

2, 2

Figure 1 The 2x2 Prisoners' Dilemma

36 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

are so willing, so that the group is 'doubly privileged', then there should
be no such presumption, unless each player is willing to contribute
regardless of what the other player does, that is, if C is a dominant
strategy for each player. If, however, each player is willing to provide
some of the public good unilaterally but not if the other player will provide
it - that is, if condition (a') fails but all the other assumptions of the
Prisoners' Dilemma game are retained - then we have a game of Chicken,
and in a game of Chicken it is not at all obvious what the outcome will be,
as we shall see.

3, 3

2,4

4, 2

1, 1

Figure 2 The 2x2 Chicken Game

This structure of preferences is more appropriate than the Prisoners'
Dilemma game as a model of certain widespread reciprocity practices
involving the production of public goods and, especially in its AT-person
generalization which we shall look at shortly, of a variety of situations
involving ecological or environmental public goods. Consider, for
example, two neighbouring cultivators whose crops depend upon proper
maintenance of dykes and ditches for flood control or irrigation. There is
a minimum amount of work which must be done; either individual alone
can do it all, but each prefers the other to do all the work. The
consequences of nobody doing the work are so disastrous that either of
them would do the work if the other did not. The structure of preferences
here is that of the game of Chicken.

Not all reciprocity or mutual aid practices resemble games of Chicken.
If the product of the reciprocal assistance is not itself a public good, the
game is more likely to be a Prisoners' Dilemma. Consider, for example,
our two neighbouring cultivators, each of whom can choose to give or
withhold assistance to the other at crucial times, such as when they need
to get a harvest in quickly. With help, each gets enough done to enjoy a
satisfactory winter; without help, a miserable winter of near-starvation
ensues. Then (if we isolate this game from any wider or continuing
relations between the two individuals) each player would prefer to have
the other help him without having to return the favour (i.e., to be a

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 37

unilateral Defector) rather than mutual assistance (since helping the
other is costly); and each would prefer mutual Defection to being a
unilateral Cooperator, since in either case he would have no help (in
getting his harvest in, etc.). Mutual Cooperation is nevertheless preferred
by both players to mutual Defection. This game is therefore once again a
Prisoners' Dilemma. It is, in fact, just an instance of exchange, in which
each party has a choice between yielding up his good or service (C) and
holding on to it (D). The asymmetric outcomes, where one player yields
and the other does not, would normally be said to involve stealing. In
anarchy, games of exchange/stealing are generally Prisoners'
Dilemmas. 5

For a second example of public goods interaction resembling a
Chicken game, consider the case of two large factories which discharge
effluent into a small lake. Each producer can choose between polluting
(D) and refraining from polluting (C). The lake can absorb waste from
one factory and still remain usable, but the wastes from both factories
carry it over a critical threshold. The resulting ecological catastrophe is
so bad that each producer, though he finds a free ride on the restraint of
the other producer preferable to mutual Cooperation, would prefer to
refrain unilaterally if the other producer pollutes (the cost of refraining
deducted from the benefits derived from this unilateral restraint being
less than the (dis)utility of the catastrophe).

We have here a case of a public good which is not provided in
smoothly increasing amounts as the level of contributions increases.
Ecological systems such as lakes, rivers, the atmosphere, fisheries and so
on can normally be exploited up to some critical level while largely
maintaining their integrity and retaining much of their use value. If
exploitation rates go beyond that critical level, use value falls cata-
strophically. With fisheries, for example, once the population has
fallen below that necessary to maintain a viable breeding stock the
species will rapidly cease to be commercially exploitable. Although the
use value of the 'common' may decline somewhat as rates of exploitation
approach the critical level, it falls catastrophically beyond that level.

A similar sort of discontinuity is found with many public goods of the
'public works' variety, such as road and rail links and bridges, which
cannot be usefully provided in any amounts but only in more or less
massive 'lumps' or tranches. In some cases no amount of the public good
can be provided until total contributions exceed some threshold. If, in

38 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

the 2x2 game, a single individual's contribution is insufficient to
provide any of the public good, or provides only a very little of it, then
each player will prefer D if the other player chooses D, but may prefer to
contribute if the other contributes too. In this case, we have a variant of a
third type of game, the game of Assurance.

C D

4, 4 1, 2

2, 1 3, 3

Figure 3 The 2x2 Assurance Game

Consider again a reciprocity practice which produces a public good :
the maintenance of dykes and ditches for irrigation or flood control by
two neighbouring cultivators. We saw earlier that if one of them alone
could do all the necessary work (though of course preferring that the
other did it all) and would do if the other did not (to avoid disaster), the
resulting game is one of Chicken. Now suppose that neither individual
can alone produce any of the public good: if the benefit of the public
good to each of them is 4 and the cost of contributing is 2, we have the
First Variant of the Assurance game shown in Figure 4. If one individual
alone can produce some of the public good (with a benefit to each of 1,
say) but not enough to justify his costs (2, again), then we get the Second
Variant of the Assurance game. For another example of this case,
consider two members of a community sharing a vegetable patch : one
individual's weeding does not keep pace with the growth of weeds,
though it enables some crop to be grown; if both weed, the crop will be
good.

We see, then, that the games of Chicken and Assurance, as well as the

C D

2, 2

-2,0

2, 2 - 1, 1

0, -2

1,-1 0,

First

Variant

Second Variant

Figure 4 Two variants of the Assurance Game

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 39

Prisoners' Dilemma, can be relevant to the problem of collective action
to provide public goods. This conclusion will be reinforced when we
come to consider the AT-person generalizations of these games and allow
for the individual's choice of contributions to vary continuously.

When individual preferences are those of an Assurance game, there is
unlikely to be a problem of collective action to provide the public good,
as there is when the game is a PD. The 2x2 Assurance game, in its
standard form or in either of the variants, has two equilibria (C, C) and
(D, D), but since both players prefer (C, C) to (D, D), neither will expect
the latter to be the outcome, so the unique Pareto-optimal outcome (C,
C) will result.

But there is an interesting collective action problem in a Chicken
game. The important feature of this game is that there are two equilibria
and in each of these one player Cooperates while the other has a 'free
ride' on the public good provided out of his contribution, so that it will
pay each player to attempt to be the first to bind himself irrevocably to
non-Cooperation, if this is feasible. This pre-commitment strategy 'forces'
the other player into Cooperation. However, where each player is able to
bind himself in this way both may realize the dangers of simultaneous
binding and may forgo this for the fully Cooperative outcome. This is,
unfortunately, unstable (at least in the single play game) so that one can
expect any Cooperation to be fragile. I shall discuss the possibility of
Cooperation in Chicken games at length below.

3, 3

1, 4

4, 2

2, 1

Figure 5 PD for Row, Chicken for Column

If Chicken and Assurance as well as Prisoners' Dilemma games can
characterize public goods interaction, then there is no reason why
hybrids of these games should not also arise. Consider for example the
game, whose payoff matrix is shown in Figure 5, in which the Row-
chooser's preferences are those of a Prisoners' Dilemma while the
Column-chooser's preferences are those of a Chicken game. Preferences
might take this form because Column (i) values the public good much

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

more highly than Row does, or can provide some amount of it at lower
cost or is better able to contribute to it than Row, or again because he
suffers much more than Row if none of the public good is provided at all
and hence is willing to provide some of the public good if Row provides
none, but (ii) does not value the public good so highly, or does not value
large amounts of it sufficiently highly that he would be prepared to
contribute more of it if Row provided some. The outcome of this game,
unlike that of the pure Chicken game, is not at all problematic. Row will
of course choose his dominant strategy D and this will make it rational
for Column to choose C, so that the outcome is (D, C). This is the only
equilibrium and it is the only Pareto-optimal strategy pair.

An N-person game of Chicken

Few interesting games in the real world (outside of international
relations) have only two players. So the TV-person generalizations of the
games considered in the last section are of greater interest than the two-
person versions. An JV-person generalization of the Prisoners' Dilemma
was discussed briefly in chapter 1. If this game is played only once, there
is no more to be said about it : universal Defection will be the outcome.
When the game is repeated, it's another story altogether, as we will see in
chapter 4 (where we will also see that the iV-person Prisoners' Dilemma
supergame can be a Chicken game). The natural way to generalize the
Assurance game is to stipulate that (i) universal Cooperation is preferred
by each player to universal Defection, and (ii) an individual will prefer C
to D if at least a certain number of other players Cooperate but otherwise
will prefer D to C, so that the only equilibria are universal Cooperation
and universal Defection, and any 'intermediate' strategy vector (in which
some players Cooperate and some Defect) will not be an equilibrium
because each player will want to change strategy either to C or to D.
Again, the analysis of this game is unproblematic : universal Cooper-
ation will be the outcome, since of the two equilibria it is preferred by
every player to the other.

The most interesting of these (one-shot) games is Chicken. Its
generalization to any (finite) number of players is less straightforward. It
seems to me that the central feature of the 2 x 2 game which should be
retained in any such generalization is the existence of an incentive for
each player to attempt to bind himself irrevocably to non-Cooperation

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

Table 1

// both the others If one Cooperates, If both the

Cooperate the other Defects others Defect

Gl : I prefer D to C C to D C to D

G2: I prefer D to C D to C C to D

G3: I prefer D to C C to D D to C

PD: I prefer D to C D to C D to C

(or at least to convince the others he is certain not to Cooperate), an
incentive deriving from his expectation that such a commitment will
compel some or all of the other players to choose Cooperation (on which
he is then able to free-ride). So we define an JV-person Chicken to be any
game having this property, with the qualification that the pre-
commitment incentive exists before the costs of commitment are taken
into account. (Pre-commitment is sometimes costly and may be so costly
as to remove the incentive.) It is of course assumed that each player
prefers universal Cooperation to universal non-Cooperation and that
each player's most preferred outcome is that he choose D while all others
choose C (this is his most profitable free ride). As it stands the definition
does not fully determine the permissible structures of preference. A three-
person case illustrates this. We know that each player prefers D to C if
both of the other players choose C. There are then four possible
preference structures. One of these is a PD (the only one in which D
dominates C) and the remaining three are labelled Gl, G2 and G3. These
four games are shown in table 1, which gives the preferences between C
and D of any one of the players (called T) in each of the three possible
contingencies (it being assumed that the game is symmetric, so that the
preference structures shown are invariant under any permutation of the
players). Each player is assumed also to prefer (C, C, C) to (D, D, D).

Now in the 2 x 2 Chicken game, each player prefers to Defect if the
other Cooperates but prefers to Cooperate if the other Defects. A natural
JV-person generalization of this is to stipulate that each player prefers to
Defect if 'enough' others Cooperate, and to Cooperate if 'too many'
others Defect. This requirement is met in the 3-person game by Gl and
G2 above; and more generally, for any number of players, the
preferences of any player must switch direction from 'D to C to 'C to D'
only once as the number of players choosing D increases. (This
corresponds to a right ward movement in a row of table 1 above.) Now in

42 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

the three games shown above, the last two columns can also be viewed as
the game between the first two players when the third (a Defecting
column-player in table 1) has already committed himself to non-
Cooperation. These two columns can now be headed by 'C and '£)'. Let
us assume that in all these 2x2 subgames (C, C) is preferred by each
player to (D, D). In Gl, then, the new row-player will choose C (since he
prefers C to D whether the column-player chooses C or D). So will the
new column-player, who has identical preferences. Thus, if one player
pre-commits himself to D, the others are certain to choose Cooperation.
The incentive for each player to pre-commit is therefore strong.

In G2, the 2x2 subgame remaining when one player has pre-
committed himself to D is a Chicken game. Since, as we shall see, there is
no obviously, unequivocally rational strategy to pursue in a Chicken
game, a row-player who commits himself to D cannot be certain that this
will compel the remaining players to Cooperate. But since there is a
reasonable likelihood that they will, so that there is some positive
incentive to pre-commit oneself to D, I shall allow that G2 is also a case
of Chicken.

In G3 the 2x2 subgame remaining when one player has pre-
committed himself to D is a game of Assurance, so the player
contemplating pre-commitment can be virtually certain that this will
compel mutual Cooperation amongst the others. In fact, for each
player in G3 D is better than C if no others Cooperate or if two others
Cooperate, but if one other Cooperates, C is the best strategy. So each
player is in two possible Cooperative coalitions and will be tempted to
try to 'force' the other two into Cooperation by pre-committing himself
to non-Cooperation. According to our central criterion, then, G3
qualifies as an iV-person Chicken game. But notice that G3 does not
have the feature, mentioned above and possessed by Gl and G2, that
each player prefers to Defect if 'enough' others Cooperate and to
Cooperate if 'too many' others Defect. Games possessing this feature
also have the property that each player has an incentive to pre-commit
himself to non-Cooperation, but the converse is not true, as G3 shows. It
is the existence of this pre-commitment incentive (if the cost of pre-
commitment is ignored) which I believe to be the distinguishing feature
of the 2x2 game of Chicken; accordingly I prefer the broader
definition of the Af-person game which admits any game having this
feature.

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 43

It is clear, then, that in the AT-person Chicken, as in the iV-person
Prisoners' Dilemma, rational individual action can lead to the unin-
tended consequence of a Pareto-inferior outcome. For in the rush to be
among the first to make a commitment to non-Cooperation (and
thereby secure a free ride on the Cooperation of others), the number so
binding themselves may exceed the maximum number of players able to
commit themselves without inducing non-provision of the public good.
Nevertheless, the prospects for Cooperation are a little more promising
in Chicken games than they are in the Prisoners' Dilemma. I return to
this later. Before doing so, let us again consider, in the light of this
account of the JV-person game, a few examples showing the relevance of
the Chicken game to practical problems of public goods provision.

Mutual aid, fisheries, and voting in committees

All three variants of the three-person Chicken game, Gl, G2 and G3, can
characterize reciprocity and environmental situations of the sort already
mentioned. Consider again the example of the irrigation and flood
control system. Suppose now that it is a public good for three cultivators,
any one of whom is able profitably to do the necessary work, the
additional benefit (from the increased provision of the public good) to
any other player who assists the first being less than the costs of such a
contribution. Again, each prefers the others to do the work, but the
consequences of nobody doing the work are so disastrous that each
would do the work if nobody else did it. The preferences here are those of
the Chicken game G2. If the job can be done profitably by one player
alone, but can be done so much better by two players that it pays a player
to contribute if another player is already doing so, then the preferences
are those of Gl. If the job cannot be done by any one of the players alone
but can be done by two of them with profit to each, then the preferences
are those of G3.

For an environmental example, consider a fishery, which can
profitably be exploited up to some critical level beyond which there is
catastrophic collapse. There are no plausible further assumptions in this
case which would yield the game G3. But suppose that the fishery can
tolerate one or two users, while three would surpass the critical level.
Fishing being costly, if two are already fishing the third would prefer to
refrain. Then (with other appropriate assumptions) the game is G2. Each

44 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

player should want to be one of the pair which can 'force' the third into
refraining from fishing. If at least two players must refrain from fishing to
prevent collapse, and the much reduced catch which a player would get if
either one or two others were fishing yields less benefit than the fishing
would cost, then the game is Gl. Each player should want to be the sole
player who can enjoy a free ride on the remaining two by pre-committing
himself to fishing and 'forcing' the others (who are then in an Assurance
game) into restraint.

This is of course a highly stylized model of a fishery, and if we were to
examine a real world 'fishery' we would find a much messier picture. In
the case of whaling, for example, I think it can be plausibly argued that,
although the 'game' once resembled a Prisoners' Dilemma, for the period
from about 1947, when the International Whaling Commission was
formed, Chicken would be a better approximation. 6 In the recent period,
which saw the elimination of all the major whaling nations except Japan
and the USSR (as well as the commercial extinction of the Blue Whale
and other commercially important species), the behaviour of Japan and
the USSR can be interpreted as attempts to demonstrate their commit-
ment to non-Cooperation with a view to forcing other whaling nations
to withdraw from the market, that is, to force them into Cooperation.

My final illustration of a public goods problem to which the game of
Chicken is appropriate is less speculative. It concerns the rationality of
the act of voting, which involves a perfectly lumpy public good.

Suppose that two options, A and B, come up before a committee
operating under simple majority rule. Attendance and voting is optional.
There is a group of players who both prefer A and know that they
collectively constitute a majority of the players. For this group obtaining
A is a pure lumpy public good (even if A itself is not a public good).
Assume for simplicity that all players outside the group (who prefer B)
turn up. So long as just enough of the /1-supporters to form a bare
majority turn up, the public good is provided; if less than this number
turn up, none of the good is provided. There is no advantage in
additional members over and above the bare majority turning up. So
long as we can assume that the advantages of A over B are greater than
the cost of voting for members of the majority group, the situation is
modelled rather well by Chicken. Each member of the group will attempt
to be among the first to find convincing reasons for being unable to
attend the committee. There are three possible outcomes : either a group

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 45

just sufficient to get A through is 'forced' into attending; or if the group
members are risk averse and the issue is vital they might all turn up,
realizing the danger that A will not be passed; or insufficient of them
attend to get A through and the public good is lost. In the first and
second situations voting is perfectly rational for those who do attend (on
expected payoff grounds in the second case).

For obvious reasons this argument makes more sense where numbers
are small (as in committees) than where they are very large (as in
constituencies).

Pre-commitment as a risky decision and the prospects for
cooperation in Chicken games

If for whatever reasons it happens that the players in a game of Chicken
are not all identically placed, so that some are unable to commit
themselves to non-Cooperation or can commit themselves only at
prohibitive cost or are unable to commit themselves as early as others,
then it may turn out that one of the stable profitable subgroups is 'forced'
into Cooperation and some of the public good is provided. Otherwise
there is, as we have seen, a danger that all the players will bind themselves
irrevocably to non-Cooperation, or that in the case of a lumpy good so
many will bind themselves that the good cannot be provided at all.
Recognizing this, risk-averse players might in fact forgo commitment
and Cooperate.

It can be argued that forgoing binding in this way is consistent with
some well-known principles of decision-making under uncertainty. The
pre-game is quite likely to be characterized by uncertainties about the
other players' attitudes towards risk and whether they were irrevocably
bound to non-Cooperation. Under uncertainty, players might choose to
maximize the minimum payoff they could get (the maximin strategy) or
choose the strategy which minimizes the difference between the best and
worst payoffs obtainable from each strategy (the minimax regret
strategy). Adoption of the maximin strategy by all players leads to the
Cooperative outcome in Chicken. Use of the minimax regret principle
can lead to Cooperative or non-Cooperative choices, depending on the
payoff differences. There are, however, well-known doubts about the use
of either of these principles, particularly in variable-sum games.

Suppose that each player is not totally uncertain about other players'

46 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

Table 2. Outcomes and fs payoffs

Less than s — 1 s — 1 others More than s - 1

others Cooperate Cooperate others Cooperate

public good

not provided

provided

-c

b-c

public good

not provided

provided

future behaviour but has rather a subjective probability which he puts on
each other player committing himself to non-Cooperation. Consider the
case of a perfectly lumpy good, which, I have argued, is often best
represented by the game of Chicken, and to simplify the analysis assume
that any cooperative coalition of at least s players out of the total of JV
can provide some amount of the public good while larger coalitions
cannot provide any more of it. (This model applies to, inter alia, the act of
voting on committees which was discussed earlier, and it is therefore not
surprising that it bears a formal resemblance to accounts of power, such
as that given by Shapley and Shubik, which relate power to the question
of whether an individual is pivotal in a committee vote.)

Suppose that a player, i, assigns a probability p to any other player
Cooperating, that Cooperation costs him c, and that the lumpy good is
worth b to him. 7 Player i faces the set of contingencies shown in table 2,
which also gives the payoff to each player for each possible outcome.
Denote by P <s - u P s - t and P >s - 1 the probabilities that fewer than s - 1,
exactly s, and more than s - 1 other players will contribute. Player fs
expected payoff if he chooses strategy C is the sum of the three payoffs in
the top row of table 2 each multiplied by the probability that the
outcome in question will occur, that is, by P< s - U P s -i and P> s -i
respectively. His expected payoff if he chooses D is calculated similarly.
Then the difference between these two expected payoffs - the expected
payoff if he chooses C less the expected payoff if he chooses D - is :

d=-c.P <s - l + (b-c)P s . 1 + (b-c-b)P >s . 1

Since the coefficients of c sum to one, this reduces to

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 47

which is of course just f s payoff if the public good is provided multiplied
by the probability that f will be pivotal in providing it, less the cost to i of
contributing. Denoting the number of coalitions of size s - 1 that can be
formed from the JV-1 other players by (?r/) in the usual way, the
expected payoff differential is

d=b( N s :[v- l (i-pr°-c

Player i will Cooperate just as long as d > 0, which implies that the
probability that exactly s-1 others Cooperate is greater than the
cost/benefit ratio c/b. If for example JV = 10, s = 3 and p=0.25, player fs
probability of being pivotal is about 0.14, so that b must be about seven
times greater than c or more if / is to decide not to commit himself to non-
Cooperation. 8

If, for given values of N, s, b and c, we plot the expected payoff
differential as a function of p, we see that it is a unimodal curve with a
maximum at p = (JV- l)/(s- 1), from which it falls monotonically on
both sides, as p decreases or increases, to a minimum of —c when p=0
and p = 1. It's clear, then, that d is negative for at least some values of p,
whatever the values of AT, s, b and c, and may be negative for all values of
these parameters. If, for example, N = 5, s = 3, c = 2 and b = 3 (i.e., the
public good is very costly to produce, or little valued by i relative to the
value of fs contribution), then there is no value of p which makes d
positive and hence f will not Cooperate whatever the subjective
probability he places on other players Cooperating. But for some values
of JV, s, b and c, the expected payoff differential d will be positive over
some (intermediate) range of values of p. Outside this range, p is either so
small that player i expects that not enough others will Cooperate for his
contribution to make the difference, or is so large that i expects so many
others to Cooperate that his contribution would be redundant. 9

This analysis of pre-commitment as a risky decision gives us a further
reason for believing that some Cooperation will be forthcoming in
Chicken games, and hence that in those public goods interactions for
which Chicken is the appropriate model some amount of the public good
will be provided. But unfortunately it has to be said that this analysis has
an unsatisfactory implication. Consider the effect on player f s decision
to Cooperate of increases in the size of the group, JV, and the size of the
smallest coalition which is able to provide the public good, s. For given
values of b and c, the behaviour of d as JV varies depends on the

48 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

behaviour of the binomial term ("lt)p s ~ Hi -pf~ s - The first part of
this, the number of subgroups of size s - 1 that can be drawn from a
group of size JV — 1, increases with N. Now suppose that player i assumes
that ceteris paribus each of the other players is less likely to Cooperate as
N increases and is more likely to Cooperate as s increases. Then the
probability that any one of the subgroups of size s - 1 will occur,
p s_1 (l— p) N s , decreases with increasing N.

It turns out that in some circumstances increasing the group size N so
increases the number of subgroups of size s - 1 in which f s Cooperation is
pivotal that this more than compensates for the smaller probability that
any such subgroup will form, with the result that i is more likely to
Cooperate as N increases. Whether or not this happens depends on how
p declines with increasing N and on the value of s - 1 relative to that of
Np. 10 In a similar way it can be shown that as s increases, the expected
payoff differential d can rise or fall (depending on the same two factors).

The upshot is that, when a player assumes every other player is less
likely to Cooperate as N (or s) increases, under certain conditions he
himself is more likely to Cooperate as N (or s) increases. We have
reached, in other words, the rather unsatisfactory conclusion that f s
behaviour as N and s change may be inconsistent with the way he
assumes others will behave. (The model is in this sense analogous to the
Cournot analysis I shall examine briefly below.)

We do not, however, encounter this problem if player i makes no
assumptions about the effects of increases in N or s on p. For some given
values of AT and s (and of b and c), there will be values of p such that his
expected payoff from Cooperation is greater than his expected payoff
from Defecting. This conclusion, taken together with the (admittedly
rather informal) points made in the first two paragraphs of this section,
give us grounds for believing that some Cooperation is more likely to
occur in games of Chicken than in Prisoners' Dilemmas, if these games
are played only once. This conclusion will be reinforced by the analysis
in the next section, where I treat games in which each player's strategy set
is continuous. But, I must reiterate, the real-world 'games' we are
concerned with are rarely played only once, and we should therefore be
more interested in the analysis of dynamic or repeated games. The
analysis of the iterated Prisoners' Dilemma is the subject of the next two
chapters. Of the dynamics of behaviour in iterated Chicken games
almost nothing is known - though an encouraging early result from

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 49

work in progress shows that, contrary to a belief popular amongst
students of international relations, it may not be rational to try to
acquire a reputation for 'toughness' (by making commitments to non-
Cooperation) if the game has more than two players. 11

Continuous strategy sets

In many (but certainly not all) cases of public goods interaction each
individual can choose to contribute a continuously variable amount
within some range, or choose from a large number of discrete amounts
which may be approximated by a continuous model.

Consider, for example, Russell Hardin's model of collective action,
introduced in chapter 1. Each individual can choose only between
contributing or not contributing one unit of the cost of producing the
good (one unit, let us say, of a numeraire private good) and every unit
contributed produces an amount of the public good with benefit r.
Hardin assumed that the benefit to an individual of the public good
produced from n units of contributions is nr/N.

Let us now modify this example by allowing each individual to choose
to contribute any amount from zero to some personal maximum. Then, if
the total contribution of all other individuals is C and his own
contribution is c, his utility is (C + c)r/N-c, so that his utility is a
linear function of c, which increases with increasing c if N < r, decreases
if N > r, and remains at a constant level if N=r. Thus, when N > r, the
game is a Prisoners' Dilemma and each individual's utility is maximized
if he chooses to contribute nothing, but when N < r he should contribute
the maximum possible.

This simple model is not at all typical of public goods interaction. In
particular, each individual's utility is a linear function of the total
amount (X, say) of public good produced and of the amount of the
private good (Y) which he contributes towards the costs of production.
Thus, his indifference curves, each one a locus of points (X, Y ) between
which he is indifferent, are linear. The transformation function, specifying
the quantity of public good which can be produced with a given input of
the private good, is also linear.

More generally, we should expect neither of these two functions to be
linear. The indifference curves normally will exhibit convexity; that is, as
the amount of either one of the goods increases, an additional unit of it

50 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

requires a smaller sacrifice of the other good in order to maintain utility
at the same level. (See, for example, the indifference curves in figure 6.)
Also, the transformation function could assume a variety of forms. Over
some range it would probably exhibit diminishing marginal returns, that
is, as the amount of public good produced increases, the cost of
producing an additional unit increases. In the case of lumpy public
goods, it would exhibit discontinuities. Let us go on to consider, then, the
more general situation in which the indifference curves have the
conventional convexity property. 12

Consider a public for which some good is perfectly indivisible and
non-excludable. Consider some member of the public, player i, who may
contribute any amount y„ between zero and some personal maximum
y„ of a numeraire private good Y, ('money'). Denote by X the amount of
public good produced, and by X the amount produced by all the
remaining players, whom I shall henceforth refer to as 'the Others'.
(Notice that X is the amount of public good, not benefit, as in the
discussion of Hardin's case above.) Assume that each individual's
preferences can be represented by the usual convex indifference curves. If
the transformation function, specifying the quantity of public good
which can be produced with a given input (y) of the private good, is linear
(i.e., an additional unit of y yields the same additional amount, r say, of X
at every level of X) and has no discontinuities, then the situation in which
i finds himself is that shown at figure 6. If the amount of the public good
produced by the Others is given and for the time being fixed, then i can
decide what is his best course of action. Suppose, for example, that the
amount of public good produced by the Others is X l . (Since the good is
perfectly indivisible and nobody is excludable, i also consumes X l ). Then
if i contributes nothing (produces no additional amount of the public
good), he is at the point A. If he devotes all of his endowment F, to the
public good, he is at point B. The points on the line AB give the whole
range of alternatives available to him, given the Others' level of
production. He will therefore choose the point P t at which his utility is
maximized - if he assumes that the Others' choices will not in turn be
influenced by his choice. Similarly, P B is his optimal response if the
Others' production of the good is zero, P 2 if it is X 2 , and so on. Clearly,
the greater the public good return on a unit outlay of the private good,
the more of the public good i will choose to produce for a given
production by the Others.

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 51

O XI Xl B X

Figure 6

It can be seen that, in this example, the more Others contribute, the
less will the individual in question contribute. In this respect fs
preferences are like those of a Chicken game.

But transformation functions are unlikely to be linear. In many cases
they will exhibit diminishing marginal returns, as in figure 7; or the
amount of public good which can be provided will at first increase only
slowly with increasing contributions, then much faster, then fall off with
diminishing marginal returns, as in figure 8. (The shape of the
transformation function facing i changes of course as Others provide
more of the public good. In effect, each transformation curve in these
figures is a lower portion of the curve to its left). If the good is lumpy, the
transformation function will be a step function, of which one plausible
form is shown in figure 9. Here, the public good cannot be provided at all

Figure 7

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 53

if total contributions are less than some threshold (y m , say), at which the
minimum 'lump' of the public good (an amount X m ) can be provided,
and beyond which increasing contributions yield diminishing returns of
public good. (Another possibility is that the public good can be provided
at only one level - a one-step pure lumpy good - in which case the
curvilinear parts of the transformation functions in figure 9 are vertical
lines.)

Figure 9 assumes that y m < Y u that is, the minimum that must be
contributed if any of the public good is to be provided is less than player
f s endowment (the most he can contribute). This ensures that 7, - y m is
positive, so that if / contributes enough he alone can cause some of the
public good to be provided. In figure 9(a) the Others' contribution (X„) is
zero. So if i contributes less than y m , none of the public good is provided,
and therefore part of the transformation function is the segment of the Y,
axis as shown. When i contributes y m or more, some amount of the public
good is provided, as shown by the curvilinear part of the transformation
function. In figure 9(b) the Others have contributed a positive amount
(y B say) but not enough to pass the threshold, so that none of the public
good is provided until i has contributed enough (y m — y ) to bring the
total contributions to the threshold y m . In figure 9(c) the Others'
contributions have reached the threshold y m exactly (the left-most
transformation curve), at which point an amount X m of the public good
is provided if i contributes nothing, or they have passed it (the remaining
curves). Note that the lumpy form shown in figure 9 is a limiting case of
the transformation curve shown in figure 8.

The earliest phase of the sequence in figure 9 has to be modified if
y m ^ 7„ that is, the minimum that must be contributed if any of the
public good is to be provided is at least as great as f s endowment. For as
long as the Others' total contribution falls short of y m by at least Y„ then
no matter how much player i contributes, he cannot reach the threshold
y m , so cannot cause any of the public good to be provided. If the Others
contribute nothing, or any amount less than or equal to y m — Y„ the
transformation function facing player i is the whole of the segment Of,
of the Y f axis. When the Others' contributions exceed y m — Y u i"s
transformation function is first as in figure 9(a), then as in figures 9(b)
and (c) as the Others' contributions increase.

Let us look now at the pattern of fs optimal responses as the Others'
contributions increase. The first point to note is that, for any of these

54 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

transformation functions, if f s indifference curves are 'sufficiently flat' -
that is, the individual's valuation of the public good is sufficiently low
relative to his valuation of his private good - then his optimal response
to any level of contribution by the Others is a zero contribution ; so that,
if this is so for every individual, the game is a Prisoners' Dilemma. (How
'flat' the indifference curves must be depends of course on the shape of
the transformation function.)

If the indifference map is not like this, then the patterns of optimal
response most likely to occur are of two kinds. The first, which is
Chicken-like, has already been encountered in figure 6 with a linear
transformation function : here, the more Others contribute, the less will i
want to contribute (and if Others contribute enough, i will want to
contribute nothing). This pattern is also illustrated in figure 7, and it is
easy to see how it could occur (with 'steeper' indifference curves) in figure
8. For the lumpy good shown in figure 9, it could arise only if the
threshold y m was sufficiently small so that a sufficiently 'steep' in-
difference curve in figure 9(a) would be tangent to the curvilinear part of
the transformation function. The second pattern is shown in figure 9 and
is likely to arise only where the transformation function corresponds to a
step or lumpy good (of which the function in figure 9 is an example) or is
of the sort shown in figure 8 (of which figure 9 is a limiting case). In this
case, fs optimal response is to contribute nothing if Others contribute
nothing (P in figure 9(a)), but to make a contribution if the Others'
contributions exceed some minimum (P u P 2 , etc. in figure 9(b) and (c)).
In this respect the resulting game (if every individual has a similar
response pattern) is Assurance-like. But when i does contribute, his
contribution declines with increases in the Others' contributions. In this
respect, the game is Chicken-like. What is happening here is that,
because of the shape of the transformation function (reflecting the fact
that little (figure 8) or none (figure 9) of the public good can be provided
out of small contributions, but beyond some threshold a small increase
in contributions yields a substantial increase in public goods provision),
an individual may find that if the Others are contributing enough (but
not too much), a small contribution from him yields so much more of the
public good that he is more than compensated for the costs of his
contribution. In figure 8, for example, if Others contribute an amount in
the region of Xq, a small contribution (y,-) from j yields a great increase
(Xi) in the public good. Similarly in figure 9(b), when Others have
contributed most of the minimum necessary (y m ) to start production of

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 55

the public good, a small additional contribution from i takes the group
over the threshold (to the point P x for example).

This Assurance /Chicken pattern is an intuitively plausible one, and
where contributions vary continuously it is likely to arise in just those
situations (discussed earlier) which would be Chicken games if the
individual could choose only between a zero contribution and a single
fixed level of contribution. In the flood control and irrigation case, for
example, if each cultivator could choose any level of contribution (work
effort) up to some maximum, then the transformation function is likely
to be of the kind shown in figures 8 or 9; so that, unless every individual
values the public good (irrigation, flood control, etc.) so little as to make
the resulting game a Prisoners' Dilemma, which is unlikely in this case,
there are two possibilities: (a) the Chicken-like case in which each
individual values the public good so much that he is able and prepared to
provide some of it when Others provide none, but will contribute less as
Others contribute more; or (b) the combined Assurance/Chicken case in
which each individual does not value the public good highly enough to
find it worthwhile to provide some of it alone, but will contribute to its
provision if Others do, though contributing less as Others contribute
more. In the irrigation and flood control example, the first of these
possibilities is perhaps less likely than the second, because although each
individual may value this public good very highly indeed he is unable
alone to provide any of it (figure 9) or enough of it to make it worth his
while (figure 8).

These are not the only possible response patterns, of course, though
they are probably the ones most likely to occur.

In chapter 1 1 suggested that indivisibility and nonrivalness should be
distinguished, though they are commonly not. The difference between
them can be clearly seen in terms of the diagrams we are using here. If a
good is not perfectly indivisible, then as group size, N, increases, the
amount available for an individual's consumption decreases. Since the
transformation function represents the consumption possibilities for an
individual, given the amounts of public good already provided by Others
and by himself, its shape would change as N varied. In fact, as N
increased in the case of an imperfectly indivisible good, the amount of the
public good available to / (for a given total contribution of the private
good) would decrease (so the transformation curve would rotate or shift
in a southwesterly direction).

On the other hand, if some rivalness was present, the indifference maps

56 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

should change as N varied, becoming 'flatter' with increasing TV to reflect
the lower value that a given quantity of the public good has to the
individual in question as more consumers are present.

The effect of an increase in N on the individual's contribution (for a
given contribution by the Others) need not be the same in the two cases.

Cournot analysis

We saw in the last section that an individual could determine his optimal
level of contribution - given the total amount of public good provided by
the remaining individuals and assuming that their choices in turn are not
influenced by his decision, that is, that each individual behaves non-
strategically (such behaviour is known as Cournot behaviour). Suppose
now that there are just two individuals and that for each of them we can
determine an optimal response to each level of public good provided by
the other. Let X° 1 (X 2 ) denote individual l's optimal response for a given
value of X 2 , the amount of the public good provided by individual 2.
Define X 2 (X i) similarly. These response functions or reaction curves can
take many forms. For the Chicken-like cases illustrated in figures 6 and 7
above, typical shapes of the reaction curves are those shown in figure 10.

X\ Xi X) rf, X t

Figure 10

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS 57

Assuming still that each individual behaves non-strategically, suppose
that initially the two individuals choose to produce amounts X\ and X 2
of the public good (the point P^ in figure 10). Then each realizes that his
own production is not optimal, given the other's choice. Individual 1 will
increase his production to X° l (X\) = X\, and individual 2 will increase
his to X 2 . This brings them to P 2 . But here, too, each has an incentive to
alter his level of production, and they will move to P 3 . This 'process' will
converge to the point P* in figure 10, the point at which the two reaction
curves intersect, and P* is the only point at which neither individual has
an incentive to change his production level unilaterally. It is, in other
words, an equilibrium. This equilibrium need not be Pareto-optimal.

On the assumption of non-strategic behaviour, an individual's
optimal response to a given level of provision of the public good by
Others - and therefore his entire reaction curve - would be the same no
matter how many other individuals there were. If we also assume that all
individuals have identical preferences and the same initial endowments
of private good that can be devoted to production of the public good,
then it is possible to examine the movement of the (Cournot) equilibrium
as the size of the group increases. This has been done by John
Chamberlin and Martin McGuire, who have shown that if the good
exhibits pure jointness or indivisibility and there is perfect nonrivalness,
then the amount contributed by each individual at the Cournot
equilibrium declines with increasing group size, tending to zero as N
approaches infinity, but that provided the public good is not an inferior
good 13 for any individual the total amount of the public good provided
at the equilibrium increases with group size. 14 (This result depends
critically on the assumption that the reaction curves do not vary with N
- which holds only if the good is purely indivisible and perfectly nonrival
and the total costs of providing the public good do not rise with N. This
second condition is not mentioned by Chamberlin or McGuire.)

If, on the other hand, the public good is not purely indivisible or there
is some degree of rivalness (though the good is still non-excludable), then
the individual's equilibrium production decreases as N increases, but the
group's total production may increase or decrease depending on how the
reaction curves vary with N. 15

These results for the case of non-strategic or Cournot behaviour are
consistent with those stated in the discussion of Olson's 'size' argument
in chapter 1. But in my view we should not attach much significance to

58 ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

them, for three reasons. First, as it is conventional to point out, the
Cournot analysis is based on the quite unacceptable assumption that
each individual reacts to what others do while assuming that they do not
react to what he does, that is, that in reacting to their choices he can
ignore the effect of his actions on theirs. Second, the analysis is entirely
static. The reactions and counter-reactions of which it speaks constitute
only a sort of pseudo-dynamics; they are merely conjectural, taking
place, as it were, only in the heads of the players. As I have argued
already, public goods provision is generally a process; interaction,
usually strategic, takes place in time. This should be modelled explicitly
(as it is in the analysis in the next two chapters). Third, in many
important public goods interactions, the reaction curves will not
resemble those shown in figure 10 and in some (perfectly plausible)
cases there will be multiple local equilibria and the Cournot analysis
will provide no reason to expect that any one of them will be the
outcome. For example, if in a two-person situation the transformation
function and indifference maps are such as to produce the As-
surance/Chicken pattern of optimal responses illustrated by figures 8
and 9 above, then the two reaction curves resemble those in figure 1 1. In
this case, the points and P* are locally stable. But starting at some
points, such as A and B in figure 1 1, the Cournot series of reactions will
not converge on or P* or on any other point.

A summary remark

The general thrust of this chapter, with or without the Cournot analysis,
has been that in public goods interaction the individuals' preferences at
any point in time are not necessarily those of a Prisoners' Dilemma
game. This is true of both two-person and Af-person games and of cases
where strategy sets are continuous as well as those where the players
have only two strategies available to them. I have argued in particular
that important classes of public goods provision problems are better
represented by Assurance and especially Chicken games and in the
continuous case by games that are Chicken-like or like a hybrid of
Assurance and Chicken. In these cases, arguably, if the games are played
only once, some cooperation is more likely to be forthcoming than in
cases for which the Prisoners' Dilemma is the appropriate model.

ALTERNATIVE GAMES IN THE PROVISION OF PUBLIC GOODS

* 2

Figure 1 1 Reaction curves for an Assurance/Chicken Game

In the next two chapters, however, I shall assume the worst: that
preferences at any point in time are indeed those of a Prisoners' Dilemma
game. But then I shall go on to consider the possibility of cooperation
when this game is repeated.

3. The two-person Prisoners' Dilemma
supergame

The treatment of the problem of public goods provision in the last
chapter was entirely static. It was concerned only with preferences at one
point in time, and conclusions about public goods provision were
derived solely from these static preferences. Individuals were supposed,
in effect, to make only one choice, once and for all (a choice of how much
to contribute to the provision of the public good). Olson's Logic of
Collective Action and other studies referred to in the first two chapters
are static in this way.

Needless to say, it is not always like this in the real world. With respect
to most public goods, the choice of whether to contribute to their
provision and of how much to contribute is a recurring choice; in some
cases it is a choice .that is permanently before the individual. This is true
of the choice of how much to exploit the 'common': how many whales to
take in each year, how much to treat industrial waste before discharging
it into the lake and so on. It is also true of the individual's choice of
whether or not to behave peaceably, to refrain from violence, robbery
and fraud and so on.

I propose to treat these recurring choices in the context of a
supergame.

Supergames

The remarks and definitions in this section apply both to the two-person
analysis to which the remainder of this chapter is devoted and to the N-
person analysis in the following chapter.

A supergame is simply a sequence of games. The games in the sequence
are called the constituent games of the supergame. In this book I shall
consider only supergames which are iterations of a single game. The

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME 61

constituent game will be a Prisoners' Dilemma in which two strategies
are available to each player: to Cooperate (C) or to Defect (£)). In each
constituent game of this Prisoners' Dilemma supergame the players
make their choices simultaneously (that is, in ignorance of the other
players' choices in that game), but they know the strategies chosen by all
the players in all previous games.

The Prisoners' Dilemma is by definition a non-cooperative game. The
Prisoners' Dilemma supergame is thus also a non-cooperative game.
Either agreements may not be made (perhaps because communication is
impossible or because the making of agreements is prohibited) or, if
agreements may be made, players are not constrained to keep them. It is
the possibility of cooperation in the absence of such constraint that I am
interested in here.

In this dynamic setting it is possible for an actor to make his choices
dependent on the earlier choices of other players. In particular,
cooperation could be made conditional on the cooperation of the other
player or players. This idea will be central in the analysis which follows.
It will emerge, as one would intuitively expect, that if cooperation is to be
sustained amongst rational egoists, it must be through the use of
conditionally cooperative strategies.

The constituent game of a supergame will be thought of as being
played at regular discrete intervals of time, or one in each time period.
Each player receives his constituent game payoff at the end of each time
period. The supergame is assumed to 'begin' at t=0 and the constituent
game payoffs to be made at t = 1, 2, 3, . . .

It is reasonable to assume that the present worth to a player of a future
payoff is less the more distant in time the payoff is to be made.
Specifically, I make the usual assumption that future payoffs are
discounted exponentially. Thus the value at time t = of a payoff X, to be
made at time t (at the end of the t ,h game) is A",aj. The number a, is called
the discount factor of player i, and its complement 1 — a, is the discount
rate. 1 It is assumed that < a t < 1. Thus the present value of a finite
payoff from a game infinitely distant in the future is zero. The discount
rates, though they may differ between individuals, are assumed to remain
constant through time. An important consequence of this assumption
(given that the constituent games do not change over time) is that for
each player the supergame in prospect looks the same at any point in
time.

62 THE TWO-PERSON PRISONERS* DILEMMA SUPERGAME

The processes in which I am interested are of indefinite length. They
are represented here by supergames which can be interpreted either as
being composed of a countably infinite number of constituent games or
as having a known and fixed probability of terminating in any time
period.

These two assumptions - of discounting and indefinite length - seem
to me to be appropriate ones for the problems I am interested in here,
and indeed for most other social and economic processes which could be
modelled by supergames. Whether or not one thinks the future should be
discounted (especially in such contexts as the conservation of non-
renewable resources or the dumping of nuclear wastes), the idea that
rational egoists playing supergames of indefinite length might actually
place as much value on a payoff to be received far into the future as on the
same payoff to be received immediately is quite implausible. 2

If the supergame has only a finite number of constituent games (and
the players know this), the 'dilemma' remains, in the sense that Defection
in every constituent game is the only undominated strategy, no matter
what the constituent game payoffs are. Consider a two-person Prisoners'
Dilemma game iterated T times. At the start of this supergame, each
player knows that in the final game (there being no possibility of
reprisals), Defection is his only undominated strategy, so he will choose
it; and he knows that for the same reason the other player will choose D.
The outcome of the final game is therefore a foregone conclusion. The
penultimate, or ( T— 1 ) th game, is now effectively the final game and the
same argument applies to it. Each player will choose D and expect the
other player to do likewise. Similarly for the (T— 2) th game, and so on,
back to the first game. 3

A supergame strategy is a sequence of strategies, one in each
constituent game. In the Prisoners' Dilemma supergame, the strategy in
which C is chosen in every constituent game will be denoted by C°° ; that
in which D is chosen in every constituent game will be denoted by
Other supergame strategies of special interest will be introduced below.
A strategy vector, in either a constituent game or in a supergame, is a list
(an ordered n-tuple) of strategies, one for each player. 4

The outcome of a constituent game is the actual state of affairs at the
end of the game. An outcome of a supergame is a sequence of outcomes,
one for each constituent game. In the constituent games and supergames
considered here, an outcome is uniquely determined by the strategies
actually chosen by the players.

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

Associated with each strategy vector in a constituent game is a payoff
vector, which is a list (an ordered n-tuple) of payoffs, one for each player.
A payoff is to be thought of as a quantity of some basic private good,
such as money, or amounts of several private goods reduced to a single
quantity of some numeraire, such as money. In this chapter and the next,
each player is assumed simply to seek to maximize his own payoff, and
his payoff scale is assumed to be cardinal (that is, it has an arbitrary zero
and unit and can be replaced by any positive linear transformation of
itself). Here the payoffs may be identified with 'utilities', in the sense that
a player (strictly) prefers one outcome to another if and only if the first
yields a greater payoff (utility) than does the second and he is indifferent
between them if and only if they yield equal payoffs. But in a later chapter
on altruism this identification is not made, for here a player's utility is
assumed to be a function of the payoffs of other players as well as his
own.

A supergame payoff to a player is the sum of an infinite series whose
terms are his payoffs in the ordinary games. The discounted value of this
payoff at t = is thus i X,a\, where X, is i's payoff at time t (his payoff
from the game in period t). Since < a, < 1, this infinite series converges
(that is, the supergame payoff is finite) for any sequence of payoffs {X,},
just as long as each X, is finite (as it always will be here). In the two-
person case, the supergame payoffs can be exhibited in a payoff matrix,
as in the ordinary game.

The concepts of dominance and Pareto-optimality have already been
introduced (in chapter 1 ). The definitions given there also apply, mutatis
mutandis, to supergames, but a few more terms are needed. The
definitions which follow apply to both ordinary games and supergames.

An outcome is said to be Pareto-preferred to another if and only if at
least one player (strictly) prefers the first to the second and no player
(strictly) prefers the second to the first.

An equilibrium is defined as a strategy vector such that no player can
obtain a larger payoff using a different strategy while the other players'
strategies remain the same. An equilibrium, then, is such that, if each
player expects it to be the outcome, he has no incentive to use a different
strategy. Thus, if indeed every player expects a certain equilibrium to be
the outcome, then it is reasonable to suppose that this equilibrium will in
fact be the outcome. But a player may have reasons for expecting that a
certain equilibrium will not be the outcome. Then he might not use his
equilibrium strategy and the equilibrium will not be the outcome.

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

This possibility is important, as we shall see, in the study of
supergames. For whereas in the Prisoners' Dilemma ordinary game
there is only one equilibrium and there is no reason for a player not to
expect it to be the outcome, in Prisoners' Dilemma supergames there are
generally several equilibria and the question arises whether some of
them may be eliminated as possible outcomes because at least one of the
players does not expect them to occur.

For convenience, the following expressions are sometimes used in this
chapter and the next. A strategy vector is said to be always an
equilibrium if and only if it is an equilibrium no matter what the ordinary
game payoffs are (as long as they satisfy the inequality which makes the
ordinary game a Prisoners' Dilemma) and no matter what values the
discount factors assume (as long as each a, satisfies < a, < 1). A
strategy vector is sometimes an equilibrium if and only if it is an
equilibrium for some but not all values of the ordinary game payoffs and
the discount factors. If a strategy vector is neither always nor sometimes
an equilibrium, then it is said to be never an equilibrium.

The general purpose of the remainder of this chapter is to study the
conditions under which cooperation of various kinds will occur in
Prisoners' Dilemma supergames. The approach will be to determine
which strategy vectors are equilibria and under what conditions, and,
where there are multiple equilibria, which of them is likely to be the
outcome of the game. The remainder of this chapter 5 will be concerned
with the two-person supergame, and the N-person game will be tackled
in the following chapter.

Unconditional Cooperation and Defection

Consider then the supergame consisting of iterations of the two-person
Prisoners' Dilemma game whose payoff matrix is : 6

X, X

w, w

where y > x > w > z. Rows are chosen by player 1, columns by player 2.
The first, and depressing, thing to note about this supergame is that it

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME 65

never pays either player to change his strategy unilaterally if both players
are playing D™, the strategy of choosing D in every constituent game
regardless of the other player's previous choices; that is, mutual
unconditional Defection is always an equilibrium. This is easily de-
monstrated. Any strategy other than D x must either result in D being
played on every move (in which case switching to it unilaterally from Z)°°
yields the same payoff) or in C being played in one or more constituent
games. Such a switch has no effect on the other player, since he Defects
unconditionally. So, in these C-moves, the player who switches gets less
(z rather than w) than he would have done if he'd stuck to D°°, while in all
the remaining moves he gets the same (w). This is so regardless of the
values of the constituent game payoffs and of the discount rates. So (D°°,
D 00 ) is always an equilibrium. It is therefore a candidate for the outcome
of the supergame. The conditions under which this disastrous result
would occur are examined below.

We note next that (C°°, C 00 ) is never an equilibrium, for either player
can obtain a greater payoff by switching unilaterally to D 00 .

In fact, any strategy pair in which either player chooses C 00 is never an
equilibrium. Against C°°, another player can always gain by changing his
strategy to D°°, since whatever he changes to will have no effect on the
first player, whose choices are unconditional. And if one player is already
using D°°, then the C 00 player can gain by switching to D°°. More
generally, any strategy which results in C being played in any constituent
game is never an equilibrium when paired with D 00 . This will include any
of the conditionally cooperative strategies to be defined below. In any
constituent game in which the strategy specifies a C move, the player
could do better by switching to D when playing against D 00 . In other
words, it doesn't pay to Cooperate against D 00 .

The possibility of conditional Cooperation

Both C and D are unconditional strategies. Since a C°° player persists in
playing C regardless of the other player's moves, he can be taken
advantage of. Suppose he instead adopts a conditionally cooperative
approach, playing C only if the other player does too. More precisely,
suppose he chooses C in the first constituent game, and in successive
games chooses C if and only if the other player chose C in the preceding
game. Call this strategy B. It has often been referred to as the Tit-for-Tat'

THE TWO-PERSON PRISONERS' DILEMMA SUPERG AME

strategy. Of course, against a player who Defects unconditionally,
playing B produces a worse outcome than would playing D°°. But if both
players use B, then mutual Cooperation is the outcome in every
constituent game. Can this be sustained; that is, is (B, B) an equilibrium?

Let us first see whether it pays either player to defect unilaterally from
B to D 00 , given that the other player is using B. The result of such a switch
is that the other player, after playing C in the first constituent game,
observes that the first player chose D in that game and therefore chooses
D in the second game, and similarly in every succeeding game. His
conditional Cooperation, in other words, 'collapses' immediately. Now
whether such a switch yields any gain depends on his discount rate, since
in making this switch the player gains in the first time period (the one in
which he switches) but loses in every period thereafter - relative, that is,
to the payoffs he would have received had he stuck to strategy B. This
would only produce a net gain if he valued later payoffs so much less than
earlier ones that the gain in the first period outweighed the losses in all
later periods. Let us derive the precise form of this condition.

If, against a player using B, player i (with discount factor a,) also uses
B, his payoff in each constituent game is x, so that his discounted
supergame payoff is the sum of the infinite series x(a i + a 2 i + a s i + . . .),
which is xa,/(l — a t ). If he switches unilaterally to D 00 , his payoff in the
first game is y and in every succeeding game w, so that his supergame
payoff is ya. + waVU — <*;)• So the switch does not yield again if and only
if this second payoff is no greater than the first. When this inequality is
rearranged, we find that we must have:

or, in terms of the discount rate,

x — w
y—w

So: if unilateral defection from (B, B) to D x is not to pay, the discount
rate must not be too great. How great depends on the constituent game
payoffs. In particular, it is intuitively obvious, and condition (3.1) shows
formally, that the smaller the 'instant' gain from defection - the difference
y — x, which we could call the player's temptation - the less likely, other
things being equal, will unilateral defection to D x yield a gain in the
supergame.

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

An Assurance game

Suppose for the moment that B and D 00 are the only strategies available
to each player or the only ones they consider, and suppose that condition
(3.1) is satisfied for both players, so that (B, B) is an equilibrium. More
strongly, suppose that the inequality in (3.1) is strict, so that (B, B) is
strictly preferred by player 1 to (D°°, B) and by player 2 to (B, Z) 00 ). Then
it is easily verified that the ordinal preferences among the four possible
outcomes are:

4,4

1,3

3, 1

2, 2

(where as usual a higher number represents a more preferred outcome).
This makes the supergame an Assurance game (as defined in chapter 2).
There are then two equilibria, (D 00 , D°°) and (B, B), but since one of them,
(B, B), is preferred by both players to the other, neither player would
expect (D 00 , D 00 ) to be the outcome, so that (B, B) would be the outcome
(because the point about an equilibrium of this kind is that i/every player
expects it to be the outcome - which means that every player is expected
to play the appropriate strategy - then it will be the outcome).

But if (B, B) is not an equilibrium, this reduced supergame is itself a
Prisoners' Dilemma, and then of course (D°°, D°°) is the outcome.

Conditions for (B, B) to be an equilibrium

We have shown that under certain conditions (B, B) is robust against
unilateral defections to £>°°. But the number of possible strategies in any
supergame is infinite. If defection by either player from (B, B) to D°° does
not pay, might it nevertheless pay to defect to some other strategy?
Consider the supergame strategy which, like B, is a tit-for-tat strategy,
but unlike B begins with D in the first constituent game. Call it B'. If one
player (i) switched from B to B' while the other (j) played B, then in the
sequence of moves which would result, (D, C) would alternate with (C,
D):

player i (B'): DC DC . . .
player ; (B): CDCD . . .

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

The same sequence would result, of course, if player i switched to the
strategy of unconditionally alternating between D and C after playing D
in the first game. Player fs discounted supergame payoff is now:

ya i + za 2 i + ya 3 i + za* + . . .

which sums to

yctj za 2 i
1-a 2 , 1-fl 2 ,

or (y + za i )a i l{\ — a 1 i ). Such a switch from (B, B) to B' would not yield
player i a gain if and only if this sum is no greater than his payoff from (B,
B), which is xaj(\ — Of). On rearranging, this condition becomes

Note that (3.2) neither entails nor is entailed by (3.1). So both
conditions must be included in the necessary conditions for (B, B) to be
an equilibrium. Or combining them, we can say that a necessary
condition for (B, B) to be an equilibrium is that both players' discount
factors, ai and a 2 , must be at least as big as the larger of the two ratios
specified in (3.1) and (3.2).

Is this a sufficient condition? It turns out that it is. At an equilibrium,
each player's strategy is the 'best response' to the other player's strategy,
in the sense that there is no strategy which it would be better to use than
the equilibrium strategy, given that the other player will use his
equilibrium strategy. Suppose that one player is using strategy B. His
first move is therefore C. Now consider the other player's response to B.

Suppose that, whatever is the best response to B, it begins with C in the
first constituent game. Then the B-player's move in the second game is C.
So if the best response to B begins with C in the first game, then it must
also play C in the second game. This is because (1) for each player the
game ahead is the same at any point in time (since both the discount rates
and the constituent game payoffs were assumed to remain constant), and
(2), a player using strategy B is responsive to the other player's choices in
the preceding game only. It follows that, if the best response to B
begins with C, it must play C in every constituent game.

Suppose now that, whatever is the best response to B, it begins with D.
(It follows that if in any later game the B-player chooses C, the best

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

response will play D in that game). Then the B-player's move in the
second game is D. There are now two possibilities for the best move in the
second game:

(i) Suppose that whatever is the best response to B, it plays C in any
game in which the B-player chooses D. Then the B-player's next
move (in the third game) is C ... so that the best response to B plays
D in this game. And so on. This generates the pattern of alternation
between (C, D) and (D, C) throughout the supergame.

(ii) Suppose instead that the best response to B would play D in any
game in which the B-player chooses D. Then the B-player's next
move (in the third game) is D. So again the best response must play D
in the third game. And so on. This generates a best response in which
D is chosen in every constituent game.

This exhausts the possibilities. Against B, the best response must be to
play C in every game (as would happen if B were played) or to alternate
between D and C beginning with D (as would happen if B' were played)
or to play D in every game. No other response can do better against B
than the best of these three. It follows that if a player cannot gain by
switching unilaterally from (B, B) to B' or D™, then no other strategy will
yield a gain. Thus, if conditions (3. 1 ) and (3.2) - which guarantee that (B,
B) is stable against defections to and B' - are satisfied, then (B, B) is
an equilibrium.

It has been shown, then, that a necessary and sufficient condition for
(B, B) to be an equilibrium is that each player's discount factor is no less
than the larger of (y - x)/{y - w) and (y-x)/{x-z).

Of course, this result does not imply that when this condition is
satisfied, (B, B) is the outcome of the game. I return to this matter below.

Axelrod's tournaments

A similar result to the one just proved can be found in Axelrod's book
The Evolution of Cooperation. His proof of necessity is essentially the one
given above, which follows the analysis in Anarchy and Cooperation.
Although in the earlier treatment (B, B) was shown to be stable against
unilateral defections to the class of strategies A k to be discussed below, as
well as defections to D°° and B', I did not prove sufficiency. Axelrod's
'proof of the sufficiency part 7 is incomplete (at least); he may have had in

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

mind the proof I gave here but this is unclear from what he actually says.

Axelrod's result actually concerns, not the (Nash) equilibrium of the
strategy pair (B, B), but the 'collective stability' of the strategy B (which
he calls tit-for-tat). The notion of collective stability derives from the
concept of an 'evolutionarily stable strategy' introduced into evol-
utionary biology by Maynard Smith, and is denned in a model which
supposes that there is a population of individuals all using a certain
strategy (S, say) and asks whether it can be 'invaded' by a single mutant
individual using some other strategy (S"). The mutant strategy is said to
invade the native strategy if it can do better playing repeatedly against a
native than a native can do against another native. Then, a strategy is
said to be collectively stable if no strategy can invade it.

Thus, a collectively stable strategy is, as Axelrod puts it, 'in Nash
equilibrium with itself, that is, if S is a collectively stable strategy then (S,
S) is an equilibrium. But of course an equilibrium need not consist of a
pair (or a population) of identical strategies (and in what follows we shall
see that pairs of different strategies can be equilibria under certain
conditions in the two-person supergame, and in the next chapter we shall
establish that there are equilibria in the AT-person supergame which are
composed of several different strategies). Nash equilibrium, then, is not
the same thing as collective stability, and Axelrod's 'Characterization
Theorem' is not very helpful in carrying out a full equilibrium analysis.

Axelrod, in fact, confines his attention to tournaments, in which
individuals play in pairs. This is true both of his theoretical analysis,
which is based, as we have seen, on the notion of a collectively stable
strategy, and of the round-robin computer tournament he conducted. In
the latter, a number of game theorists, economists, etc., submitted
strategies, each of which was paired against itself and against each of the
others in a Prisoners' Dilemma supergame. The strategies were then
ranked according to their aggregate performance, (tit-for-tat - the
strategy I have called B - won, and in a second competition, involving
strategies submitted in the light of the results of the first competition, it
won again.)

Axelrod comes to the same general conclusion we arrive at here (and
which was at the heart of the analysis of the Prisoners' Dilemma
supergame in Anarchy and Cooperation), namely that 'the two key
requisites for cooperation to thrive are that the cooperation be based on
reciprocity, and that the shadow of the future is important enough to
make this reciprocity stable.' 8

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

But is his approach, based on the idea of the tournament and the
concept of a collectively stable strategy, to be preferred to an analysis
using the notion of a (Nash) equilibrium? I think not. Pairwise
interaction may be characteristic of non-human populations (though
even this is doubted by Maynard Smith himself), but it certainly does not
characterize most human interactions which give rise to collective action
problems. These of course generally involve more than two individuals
and, especially where the provision of public goods is at stake, an
individual's behaviour typically depends on the whole aggregate pattern
of behaviours of the rest of the group. For example, his decision whether
or not to cooperate in the provision of a public good would generally be
contingent on there being enough others cooperating. Situations of this
kind cannot be characterized in terms of pairwise interactions. Even
where, in the real world, interactions are truly pairwise, they are most
unlikely to take the strange form assumed by Axelrod: his analysis
hinges on the assumption that an individual will play out the whole of an
infinite supergame with one other player, or each player in turn, rather
than, say, ranging through the population, or part of it, playing against
different players at different times in the supergame (possibly playing
each of them a random number of times). 9

Axelrod, admitting that his book 'will examine interactions between
just two players at a time', suggests that 'situations that involve more
than pairwise interaction can be modelled with the more complex n-
person Prisoner's Dilemma', 10 but does not attempt the analysis himself.
We shall take up the study of N -person Prisoners' Dilemmas in the next
chapter.

Coordination equilibria

It is worth mentioning in passing that if(B, B) is an equilibrium in the
two-person Prisoners' Dilemma supergame it is also a coordination
equilibrium. A coordination equilibrium is a strategy pair such that either
player is made no better off, not only if he himself unilaterally changes
strategy, but also if the other player changes strategy. Consider (B, B). In
the case of the other player defecting to D x , the effect on the non-
defecting player is the same as it would have been if he had defected,
except in the first constituent game, where now he gets even less than he
would have done had he defected (z rather than y); so that if his own
defection makes him no better off, the other's defection certainly will not.

72 THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

Similar remarks apply to changes to other strategies (except where such
a change produces the same sequence of constituent game outcomes):
for example, if defection to B' by either player makes him no better off,
then nor will he be made better off by the other player's defection to B',
since the two resulting sequences of constituent game outcomes are
identical - alternation between (C, D) and (D, C) - except that the
sequence begins with (D, C) if he defects and (C, D), which is worse for
him, if the other defects. Thus, if (B, B) is an equilibrium, it is a
coordination equilibrium. This is not, of course, true of (Z) x , D°°), which
is always an equilibrium but never a coordination equilibrium.

Russell Hardin believes that the fact that (B, B) and other strategy
vectors are coordination equilibria makes an important difference to the
explanation of behaviour in Prisoners' Dilemma games. A coordination
equilibrium, he says, is even more likely to be the outcome than a mere
equilibrium because it is 'supported by a double incentive to each
player', for each player has an interest in himself conforming with the
equilibrium and an interest in the other player conforming as well. 11

Actually, as we have seen, in the Prisoners' Dilemma supergame with
discounting, (B, B) need not even be an equilibrium. But we have to
assume that Hardin has in mind an indefinitely iterated Prisoners'
Dilemma with no discounting, since strangely he nowhere mentions
discounting, and that each player's supergame payoffs are long-run
averages of the constituent game payoffs. As I have already argued, these
games are of little importance in the analysis of social life. They are also
much simpler analytically, since they involve none of the complex trade-
offs between payoffs in different time periods that are possible in
supergames with discounting. In these no-discount games, (B, B) is in
fact always an equilibrium and also a coordination equilibrium.

But even if (B, B), or any other strategy vector, is a coordination
equilibrium, this fact does not provide each player with a 'double
incentive' to conform to it. That / want the other player to conform is of
no relevance to him, for we have assumed that he, like me, is a rational
egoist: my interest in his conforming has no effect on his actions, just as
my actions are unaffected by his desire that I should conform. The
possibility that my interest in his actions would lead me to do something
to ensure that he acts in the right way is not one that can be considered
within Hardin's framework (or mine). 'My interest in your conforming

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

means that, if there is a way to do so at little or no net cost to me, I will
want to give you further incentive to conform', says Hardin, 12 but
options in this cost-benefit comparison are not in the model to begin
with and, as always, should not be wheeled in ad hoc.

The existence of a coordination equilibrium, then, does not give a
player two reasons for conforming to it, and the fact that an equilibrium
is a coordination equilibrium does not make it doubly likely to be the
outcome. I take up the (complicated) question of which equilibrium will
be the outcome in a later section.

Other mutual Cooperation equilibria

The tit-for-tat strategy is not the only strategy which, if used by both
players, will sustain mutual Cooperation throughout the supergame.
Any strategy which plays C on the first move and then continues to play
C if the other player does, will do. Any two (possibly different) strategies
of this sort will sustain such cooperation regardless of what the strategies
require each player to do when the other player defects. Call such a
strategy conditionally cooperative. The conditions for a pair of such
strategies to be an equilibrium will depend on the particular strategies
chosen. Consider for example the class of strategies A k discussed by
Shubik. 13 A k , where k is a strictly positive integer, is defined as follows:

C is chosen in the first game, and it is chosen in each subsequent
game as long as the other player chooses C in the previous
game; if the other Defects in any game, D is chosen for the next k
games; C is then chosen no matter what the other player's last
choice is; it continues to be chosen as long as the other player
chooses C in the preceding game; when the other player next
Defects, D is chosen for k + 1 games; and so on; the number of
games in which the other player is 'punished' for a Defection
increases by one each time; and each time there is a return to C.

Denote the limiting case of A k when k oo by A w : in this case, C is
chosen until the other player first Defects, after which D is chosen in all
succeeding constituent games.

These strategies are special cases of a class of strategies I will label
A kM where k is a strictly positive integer and / is a non-negative integer.

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

A k j is the same as A k except that the 'punishment' period (which is again
of k moves duration after the other player's first Defection) is lengthened
by / Defections after each succeeding Defection by the other player.
When / = 0, the punishment periods are all of the same length. When
/= 1, A kJ is equivalent to A k .

The conditions for the strategy pair (,4*,,, A Kl ) to be an equilibrium
are easily derived. If one player (i) defects unilaterally from this strategy
pair to D 00 , then the other player (j) will choose C in game 1 (t = l),
followed by D for the next k moves, then C at t = k + 2, followed by D for
the next k + l moves, then C at t = 2k + l + 3, and so on. So fs total
discounted payoff from the constituent games in which j plays C (and i of
course plays D) is :

y(a i + a k i + 2 +a 2 i k+, + 3 +. . .)

The infinite series in parentheses is obviously convergent, since it is less
than the convergent series ^ + 0] + ^ + . . . Call its sum S(k, I; a,-) or S t
for short. We shall not need to find S, in closed form.

Player i's discounted payoff from all the remaining moves (in which
both players choose D) is then w[a,/(l -a.-J-SJ. Hence i's total
discounted payoff from the supergame is:

(y-w)S i + wa i /(l-a i )

Player i does not, therefore, gain by defecting to from (A kJ , A kil ) if
and only if this sum is not greater than the payoff from mutual
Cooperation throughout the supergame, which is xa,/(l -a,). This
condition, on rearranging, is:

\ "i J y-

Obviously, if mutual defection to D°° from (A u , A k J ) does not pay, then
defection to D°° from (B, B) will not pay - since the former, unlike the
latter, benefits from the other player's periodical return to playing C. So
if condition (3.3) is satisfied, condition (3.1) should be satisfied too. This
is the case, since S, ^ a, regardless of the values of k and /. Note that as
k -* 00 (and hence the strategy A kA becomes A x ), S, -> a h so that as
k -» 00 condition (3.3) becomes condition (3.1).

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

If player i defects unilaterally from (A K ,, A K ,) to B', the sequence of
moves is:

i (B') \ DCD . . . DDCD . . . DCD . . .
j (A k ,): CDD . . . DCDD . . . CDD . . .

I 1

t= 123 . . . .k + 2 . . . .2k + l + 3 . . . .

Comparing this sequence of moves with that of D°° against A ki ,
considered above, it is clear that player i's payoff gain from moving to B'
is less than his payoff gain from moving to D 00 . Hence if (3.3) is satisfied
then (A k<h A kyl ) is also stable against unilateral defections to B'. And,
more generally, against a player (J) who sticks to A kih any switch by
player i from A k>l to a strategy which includes C-moves during the
punishment period (throughout which, remember, the other player
chooses D regardless of i's moves, apart of course from the Defection by i
which triggers the punishment) could do no better than a strategy which
plays D throughout the punishment period, but is otherwise the same.
The only strategy which might then improve on when played against
A k _ b is one which plays C in the game in which j returns to playing C at
the end of each punishment period. But if a strategy must play C at that
point in order to be a better reply to A Kl than D°°, then it must play C at
the first move. Hence, against A kJ , no strategy can do better than the
better of D°° and any conditionally cooperative strategy. Thus, condition
(3.3), for both players, is a necessary and sufficient condition for (A kM
A k j) to be an equilibrium.

As the initial 'punishment' period or the additional 'punishment' per-
iods lengthen, that is, as k and / increase, so S, decreases. It follows that,
for given values of x, y, w and a,, condition (3.3) is 'increasingly likely' to
be satisfied as k and/or / increases. In other words, a longer 'punishment'
period (a bigger value of k or Z) will succeed in deterring a player from
switching to D°° (and therefore from switching to any other strategy)
where a shorter 'punishment' period has failed. Some economists,
treating general non-cooperative supergames, have restricted their
attention to the analogue of the iV-person generalization of the strategy
A,,., in which eternal 'punishment' (D in every succeeding game) is
triggered by a single Defection by the other player. 14 1 agree with Shubik
that this strategy contains an 'implausible' threat, and that the threats
embedded in strategies A k , for finite k, are more plausible. 15 But then, it

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

seems to me, the strategy B (which Shubik does not mention) is even
more plausible than any A k .

Taking it in turns to Cooperate

There are many equilibria in the supergame besides those examined so
far. It would be convenient if all the possible equilibria were (like those
considered so far) either such as to result in mutual Cooperation
throughout the supergame or such as to result in mutual Defection
throughout. Unfortunately, this is not the case. One which does not fall
into either category is of special interest, especially (as we shall see later)
as it may be preferred by both players to mutual Cooperation
throughout the supergame. This is the strategy pair in which one player
uses B (tit-for-tat starting with C) and the other player uses B' (tit-for-tat
starting with D). Earlier, in discussing the stability of (B, B), we saw that
the pair (B, B') produces an alternation of (C, D) and (D, C) throughout
the supergame. Under what conditions is it an equilibrium?

Suppose the B player is i and the B' player is Before either player
changes strategy, i's payoff is (z + ya^a,/^ — a 2 ,) and /s payoff is
(y + zaj)aj/(l -a)). It is easily verified that a change by i from B to C°°
yields no gain if and only if

v—x

a, < - (3.4)

x — z

This is the reverse (not the negation) of condition (3.2).

A change by i from B to £>°° or to B' produces mutual Defection
throughout the supergame and yields i no gain if and only if

w — z

a, > (3.5)

y — w

It also turns out that a change of strategy by player j from B' to B or
C 33 yields j no gain if and only if condition (3.4) holds, with j replacing i of
course; and a change of strategy by from B' to Z)°° yields j no gain if and
only if condition (3.5) holds for

Thus conditions (3.4) and (3.5) are both necessary for (B, B') or (B', B)
to be an equilibrium.

We can now show that these conditions are also jointly necessary and
sufficient by applying the argument that was used earlier in proving the

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

sufficiency of the condition for (B, B) to be an equilibrium. The argument
there showed that, against B, the best response must be to play C in every
constituent game or to play D in every game or to alternate between C
and D beginning with D (as would happen if B' were played). It follows
that if it does not pay to switch unilaterally from B' to C 00 or £>°°, when
the other player is using B, then it does not pay to switch to any other
strategy (including, for example, any strategy in the class A kil ). A strictly
analogous argument establishes that, against B', the best response must
be to play C in every constituent game or to play D in every game or to
alternate between C and D beginning with C. And from this it follows
that if it does not pay to switch unilaterally from B to C 00 or D 00 or B',
when the other player is using B', then it does not pay to switch to any
other strategy.

Thus, conditions (3.4) and (3.5), each holding for both players, are
necessary and sufficient conditions for (B, B') or (B', B) to be an
equilibrium.

Putting the two conditions together, the necessary and sufficient
condition is:

w—z y—x

< a,- <

y—w x—z

It is easily checked that this can be satisfied for some but not all
permissible values of x, y, z, w and a,.

So the pattern of alternation, in which the players take it in turns to
Cooperate, can be an equilibrium. Whether it will ever actually be the
outcome is another matter, which I address in the next section.

Before doing so, let us note the necessary and sufficient conditions,
which the reader can easily derive, for (B', B'), (B', D ») and (D°°, B') to be
equilibria. Each of these results in mutual Defection throughout the
supergame. For (B', B') the equilibrium conditions are: the reverse (not
the negation) of condition (3.5) above (for both players) together with
the condition

x — z

for both players. For (B', D°°) the equilibrium conditions are (3.6) and
the reverse of (3.5), for i = 2 in each case. For (D°°, B') the conditions are
again (3.6) and the reverse of (3.5) but now for i = 1 in each case.

78 THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

Table 3

(3.1) & (3.2)
for i = l, 2

(3.5) & rev (3.2)
for i=l, 2

Never an
equilibrium

(3.5) & rev (3.2)
for i=l, 2

(3.6) & rev (3.5)
for i'=l, 2

(3.6) & rev (3.5)
for i = 2

D x

Never an
equilibrium

(3.6) & rev (3.5)
for i = l

Always an
equilibrium

Outcomes

We have not so far considered every possible pair of strategies; nor do I
intend to try. So that, even though we have established necessary and
sufficient conditions for those we have considered, the following
discussion, which attempts to indicate which of these equilibria is likely
to be the outcome of the supergame, is incomplete. I shall confine the
discussion to strategy pairs formed from B, B' and D°°. It is unlikely that
an equilibrium which is not examined here would be the outcome, since
it would be equivalent to or Pareto-dominated by at least one of those
that are considered. Table 3 assembles the relevant conditions.

The seven possible equilibria here give rise to just three distinct
patterns of choices in the supergame: mutual Cooperation throughout,
which occurs in the case of (B, B); the alternation pattern or 'taking it in
turns to Cooperate', which results from (B, B') and (B\ B); and mutual
Defection thoughout, which results from Z)°°) and the remaining
three equilibria.

We noted earlier that an equilibrium is such that, if every player
expects it to be the outcome, and therefore expects all the other players to
choose the strategies appropriate to this equilibrium, he has no incentive
to choose other than his equilibrium strategy; so the equilibrium will be
the outcome. If a game has only one equilibrium, every player would
expect it to be the outcome, so it would in fact be the outcome. If, for
example, (D x , is the only equilibrium in this two-person Prisoners'
Dilemma supergame (i.e., none of the conditions for any of the other
strategy pairs to be equilibria are satisfied), then it will be the outcome.
But if two or more equilibria occurred simultaneously, then the fact that
a certain strategy pair is an equilibrium is not in itself a sufficient reason

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

for any player to expect it to be the outcome. But if, of two
simultaneously occurring equilibria, one was preferred by both players
to the other, then presumably no player would expect the second to be
the outcome, so it would not be the outcome.

So it would be most convenient if, whenever several equilibria occur
simultaneously, one of them was preferred by each player to all the
remainder (or at least was Pareto-preferred to them, i.e., was no less
preferred by either of the players and was strictly preferred by at least one
of them). Then we could say that no player would expect any of the
Pareto-dominated equilibria to be the outcome and it would not be the
outcome. Suppose, for example, that (B, B) is the only equilibrium
besides (D°°, D°°), which is always an equilibrium. (B, B) is of course
strictly preferred by both players to (D 00 , D°°), which nobody, therefore,
would expect to be the outcome. So (B, B) will be the outcome. (The same
conclusion, incidentally, would be reached if A kJ were added to table 3.
Any or all of the four pairs formed from B and A kJ can be equilibria, but
none of the pairs formed from A u and either B' or D can be. All four, of
course, lead to the same outcome - mutual Cooperation throughout the
supergame.) Unfortunately, such a straightforward relation of Pareto-
dominance between coexisting equilibria does not always obtain.

Suppose, for example, that the two alternation pairs (B, B') and (B', B)
are the only equilibria besides (D x , D 00 ). Then it follows that each of the
alternation pairs is Pareto-preferred to the mutual Defection equilibrium.
For if (as required by equilibrium) the B player in either of the
alternation equilibria does not prefer to defect unilaterally to D x (which
would result in mutual Defection throughout), then he must prefer the
alternation equilibrium to (£)°°, D°°) or be indifferent between them; and
if the B' player does not prefer to defect unilaterally to £>°° (as required
by equilibrium), then he certainly strictly prefers the alternation pattern
to {D x , D x ), because the pattern resulting from such a defection is the
same as the (Z) x , D 00 ) pattern, except in the first game where he is better
off (as unilateral Defector) in the former than in the latter. Thus,
whenever (B, B') or (B', B) are equilibria, each is Pareto-preferred to
(D°°, £> x ). In this case neither player will expect (D™, D x ) to be the
outcome and it will not be the outcome. This still leaves two equilibria
and between these, unfortunately, the two players have opposed
preferences : player 1 prefers (B', B), in which he Defects first, to (B, B') in
which player 2 Defects first, and player 2 has the opposite preference. So

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

even if both players eliminate from consideration all strategies except B
and B', they still face the 'coordination' problem of avoiding (B, B) and
(B', B') as well as the problem posed by their conflicting preferences
between the two equilibria. Within the formal framework we are using,
there is no resolution of this problem. In a richer specification of the
model, which could be made in any particular application, a solution
would no doubt be indicated. The problem has features in common with
the problem of Chicken, discussed in the last chapter, and some of the
remarks made in the earlier discussion apply here also. But unlike
Chicken players, both players here prefer either of the asymmetric
equilibria to mutual Cooperation. One of the alternation patterns is
therefore likely to be the outcome; but which one we cannot say.

So far, then, we have seen that there can be three distinct types of
outcome to the supergame:

(i) mutual Defection throughout: this occurs when (D x , £>°°) is the
only equilibrium;

(ii) mutual Cooperation throughout : this occurs when (B, B) is the only
equilibrium besides (D 00 , £>°°);

(iii) alternation between (C, D) and (D, C): this occurs when (B, B') and
(B', B) are the only equilibria besides (D 00 , D°°).

This conclusion is not modified when other equilibria in table 3 coexist
with these.

It is exceedingly unlikely that (B, B) is an equilibrium as well as (B, B')
and (B', B). For then both condition (3.2) and its reverse must hold,
which requires that each player's discount factor is exactly
(y — x)l(x — z). I think we can ignore this possibility; but if it did occur,
then the B' player (in each of the alternation equilibria) would be
indifferent between alternation and mutual Cooperation, and which of
these outcomes prevailed would simply depend on the other player's
preferences between them. 16

The same is true when in addition to these three equilibria (B', B') is
also an equilibrium. For then not only must condition (3.2) and its
reverse both hold, but also condition (3.5) and its reverse. So we must
have a, = (y — x )/(x — z) and a,- = ( w — z)/(y — w) for both players. But then,
in any case, both players are indifferent between B and B', for both yield
the same payoff no matter whether the other player chooses B or B'. An
even more remote possibility is that all seven of the possible equilibria in :

THE TWO-PERSON PRISONERS' DILEMMA SUPERGAME

table 3 are simultaneously equilibria, for then conditions (3.2) and (3.5)
and their reverses must hold as well as conditions (3. 1 ) and (3.6) for both
players. This is possible, but very unlikely. (E.g., set y=3, x = 2, w= 1,
z=0. Then a, must be exactly 0.5 for both players.)

If any or all of (B', B'), (B', D°°) and (D°°, B') are equilibria
simultaneously with (D 00 , D x ), then if no other strategy pairs are
equilibria it does not matter which of B' and D°° each player chooses.
Mutual Defection throughout the supergame will be the upshot in any
case. If these four equilibria occur simultaneously with those in the
second and third cases considered above, then the conclusions in each
case are unaffected : if (B, B) is also an equilibrium then it is the outcome;
and if the alternation pairs are also equilibria, then one of them is the
outcome.

One thing that emerges clearly from this analysis, then, is that if each
player's discount rate is sufficiently low, the outcome will be mutual
Cooperation throughout the supergame. For if the discount factors are
both greater than (j-x)/(x-z), then (B, B') cannot be an equilibrium
(see condition (3.4)); and if both factors are also greater than
(y-x)/{y — w), then (B, B) is an equilibrium; and if (B, B) is an
equilibrium then it is the outcome.

4. The iV-person Prisoners' Dilemma
supergame

In this chapter I take up the analysis of supergames whose constituent
games are Prisoners' Dilemmas with any finite number (JV) of players. It
should hardly need emphasizing that in the real world most of the
interesting problems of public goods provision, and of cooperation more
generally, involve more than two actors, so that much more attention
should be directed at the analysis of N-person supergames than of two-
person supergames. Yet very little has been written about such games.
Most of what has been done establishes only the existence of equilibrium
in general classes of supergames (and, as we have seen, Axelrod's analysis
is confined to 'tournaments', in which the interactions are all pairwise).
In some respects, results from the two-person analysis generalize in a
relatively straightforward way to the iV-person games. But new and
thorny problems emerge. In particular, as we shall see, in those cases
where subsets of the players find it collectively worthwhile to provide the
public good, there arises a quite different strategic problem, which results
from some players having an incentive to ensure that the subset which
provides the public good does not include themselves. In this way, a
Chicken-like game is generated by the Prisoners' Dilemma. 1

The analysis will be far from exhaustive. I shall, however, take the
discussion far enough to show how Cooperation can arise in the N-
person Prisoners' Dilemma, no matter how many players there are, and
furthermore can be sustained amongst a subset of the players in the face
of unconditional Defection throughout the supergame by the remaining
players.

The penultimate section will give an informal summary and com-
mentary on the main results and will look at what the formal analysis
tells us about the empirical conditions under which cooperation is likely
to occur. The reader who does not stay with the necessarily involved

THE A/-PERSON PRISONERS' DILEMMA SUPERGAME

(though hardly at all mathematical) analysis which follows can turn to
this section directly.

Payoffs in the constituent game

As in the two-person analysis, the supergame considered here consists of
a countably infinite number of iterations of a single constituent game.
The payoffs associated with a given outcome of the constituent game,
then, do not change from game to game. This rules out the possibility
that the payoffs in the constituent game are functions not only of the
players' choices in that game but also of their choices in previous games.
(I comment briefly in the final section on the effect of relaxing this
assumption.) As before, the constituent games will be thought of as being
played at regular discrete intervals of time, or one in each time period.
The supergame is assumed to 'begin' at time t=0 and the constituent
game payoffs to be made at t= 1, 2, 3, . . .

I assume that a player's payoffs in each constituent game depend upon
two things only: his own strategy (C or D) in that constituent game, and
the number of other players choosing C in that game. The second part of
this assumption is in fact very weak. It is equivalent to assuming that
payoffs are independent of the labelling of the players. Its relaxation
entails that payoffs depend upon which other players choose C.

Denote by/(v) the constituent game payoff to any player when he
chooses C and v others choose C, and by g(v) the payoff to any player
when he chooses D and v others choose C. I assume that f(v) and g(v) are
the same for all players.

The following three assumptions about the functions / and g seem
appropriate for the class of problems of interest here:

(i) g(v) > f(v) for each value of v ^ 0.

(ii) /(N-l)>0(O).

(iii) g(v) > 0(0) for all v > 0.

The first assumption is true if and only if D dominates C for each player:
no matter what strategies the other players use each player prefers D to
C. The second assumption is true if and only if each player prefers the
outcome which occurs when all players Cooperate to that which occurs
when all the players Defect. Thus (i) and (ii) are necessary and sufficient
conditions for the constituent game to be an N-person Prisoners'

84 THE N-PERSON PRISONERS' DILEMMA SUPERGAME

Dilemma (according to the definition given in chapter 1). Assumption
(iii) is eminently reasonable; in fact, a much stronger assumption usually
holds in practice, namely that both f(v) and g(v) are strictly increasing
with v. 2

Unconditional Cooperation and Defection

C 00 and D 00 will again be used to denote the supergame strategies of
unconditional Cooperation and Defection throughout the supergame. A
player using C°°, then, chooses C in every constituent game regardless of
the choices of all the other players in the preceding game.

As in the two-person case, universal unconditional Defection - the
strategy vector (D°°, . . ., D 00 ) - is always an equilibrium. For any
strategy vector other than must either result in D being played on
every move (so that there is no profit in switching to it unilaterally) or in
C being played on at least one move. Since a change of strategy, of any
kind, by any of the players, has no effect on the subsequent behaviour of
any of the D°° players, the player who switches must get less -f{0) rather
than #(0) - on each of these C-moves than he would have done had he
stuck to So (D°°, Z) 00 , . . ., D°°) is always an equilibrium.

Clearly, universal unconditional Cooperation - the strategy vector
(C°°, C°°, . . ., C°°) - is never an equilibrium. Any player gains by
switching unilaterally to D°°, which yields an increase in payoff from
f{N — 1 ) to g(N — 1 ) in every constituent game. In the two-person case, if
any player uses C°°, or if any player when paired with a D 00 -player uses a
strategy which results in C being played in any constituent game, the
strategy vector cannot be an equilibrium. These propositions do not
generalize to the N-person game, where, as we shall see, strategy vectors
can be in equilibrium even though some of the players use strategies of
unconditional Cooperation or Defection. But it is the case, of course,
that any strategy vector in which some players use C°° and all the rest use
D 00 is never an equilibrium, for any of the C 00 players could gain by
switching unilaterally to D 00 .

Conditional Cooperation

It is intuitively clear that, as in the two-person supergame, if any
cooperation is to be sustained at all, some of the players must use

THE JV-PERSON PRISONERS' DILEMMA SUPERGAME

conditionally Cooperative strategies. Consider first a natural generaliza-
tion of the tit-for-tat strategy B, which I will call B„. A player using B n
chooses C in the first constituent game, and thereafter chooses C in any
game if and only if at least n other players chose C in the preceding game.
I assume henceforth that n > (when n=0, B„ degenerates into C°°).
When n = N-l, a B„-player's Cooperation is conditional on the
Cooperation of all the other players.

If every player uses B m then universal Cooperation is sustained
throughout the supergame. Are there conditions in which the strategy
vector (B m B m . . ., B n ) is an equilibrium?

Suppose to begin with that n is the same for every player. I first show
that if n = N - 1 the strategy vector (B„, B n , . . ., B n ) can be an equilibrium.
In this special case, there are exactly enough players choosing C in each
constituent game to maintain Cooperation by every player in the following
game. If any player at any time chooses D, then all the remaining players
cease to Cooperate in the succeeding game. We might say there is a tacit
'compact' amongst all the players which breaks down as soon as any one
of them once breaks ranks. Suppose some player, i, unilaterally changes
strategy to D. Then in the constituent game in which he first Defects, he is
the only non-Cooperator and his payoff is therefore g{N - 1); but in all
succeeding games everyone Defects and i's payoff is therefore 0(0). It is
obvious that this defection to £>°° will only yield i a gain if the discount
rate is sufficiently high, so that his (slightly discounted) immediate gain
from free-riding on the Cooperation of all the other players outweighs
the (increasingly discounted) losses from the subsequent string of
universal Defections.

In fact, fs payoff from universal Cooperation throughout the
supergame is f(N- l)a,/(l — a t ) and his payoff if he changes his strategy
to D 00 is g(N - 1 )a, + g(0)a 2 i/(l - a,), so that the defection does not yield a
gain if and only if the first of these is no less than the second. On
rearranging, this condition becomes:

„ ^ g(N-l)-f(N-l)

0i> g(N-l)-g(0) < 4 - J >

This is the N-person analogue of inequality (3.1) in the two-person
analysis.

Let us now introduce the strategy B„ which is the AT-person

86 THE N-PERSON PRISONERS' DILEMMA SUPERGAME

generalization of B'. It is the same tit-for-tat strategy as B„, except that D
is chosen in the first constituent game.

It is easy to see (and here already the AT-person case differs from the
two-person) that if (B n , B n , . . ., B„) is stable against unilateral switches to
D 00 , then it is stable against switches to B' m if n is still AT - 1. Following
such a switch, in the second constituent game each of the N - 1 players
still using B„ will find that only N — 2 others played C in the first game, so
will choose D. The player who switched to B' n will however Cooperate in
the second game since there were N-l others (the B„ players) who chose
C in the first game. But thereafter everyone will choose D. So the
outcomes here differ from those which resulted from one player's switch
to D x in the second constituent game only, where the defector (playing C
while the rest play D) receives less than does the defector to D 00 (who plays
D while the rest play D). So if defection to D°° does not pay, defection to
B' n will not either.

In fact, it is easily shown that no strategy does better than B„ and D 00
when all others play B n and that therefore inequality (4.1), for every
player, is a necessary and sufficient condition for (B„, B„, . . ., B„) to be an
equilibrium, assuming that n = N -lfor all players. For if a strategy is to
do better than B„ when all others are playing B„ it must at some stage
(and therefore at the first move) play D. But this causes all the others to
Defect in the second game, and no matter what the switcher does in this
(second) game, all the others will defect in the third game and therefore in
all succeeding games. So in the second game (and in each of the following
games) the switcher can do no better than to play D. Hence no strategy
can do better than D 00 or B„ when playing against N-l B„-players, with
n = N-l.

Can the strategy vector still be an equilibrium if n # N — 1 for some or
all of the players - that is, if the Cooperation of one or more players is
conditional on the Cooperation of fewer than all the other players?

If the Cooperation of all N players is conditional on fewer than N - 1
others Cooperating, then universal Cooperation is sustained through-
out the supergame as before, but now, if any single player changes
strategy from B„ to D x it has no effect on the subsequent moves of any of
the other players: for each of them there are N — 2 other players
Cooperating and their own Cooperation requires only that number of
Cooperators at most. After the defection, therefore, they continue to
Cooperate throughout the supergame. Hence, from the second game

THE JV-PERSON PRISONERS' DILEMMA SUPERGAME 87

onwards the player switching to D x receives the maximum payoff - the
lone Defector's payoff g(N— 1). It therefore always pays him to switch,
regardless of his discount rate. So (B„, B m . . .,B„) cannot be an equilibrium
if n < N — l for each player.

This is still the case when the values of n vary between the players, just
as long as all of them (or all but one of them) are smaller than N-l. For,
if all of them are smaller than N-l, then after a player defects to D 30 , all
the other players continue to Cooperate (while if the values of n are less
than N — 1 for all but one of the players and this player switches to D, the
remaining players continue to Cooperate).

If the value of n varies among these B n players but for more than one of
them n = N — l, then a variety of patterns of choices could follow a switch
by one player to D°°. It is possible that an immediate but once-for-all
partial collapse of Cooperation would ensue, or that there would be a
progressive collapse, perhaps with an additional player or a small
number of players dropping out of Cooperation in successive games. It is
clear that, in some cases of this kind, a change from B„ to D°° would not
yield any player a gain, so that the strategy vector in question would be
an equilibrium, provided (as usual) that the relevant discount rates were
not too great. The conditions which the discount rates would have to
satisfy would depend of course on the distribution of the values of n
among the players, which determines the particular pattern of choices in
the constituent games.

So far, then, we have shown that the strategy vector (B„, B n , . . ., B n ), in
which each player Cooperates conditionally on the Cooperation of every
other player, is an equilibrium if and only if the inequality (4.1) is satisfied
by every player's discount rate. This strategy vector can still be an
equilibrium if some of the players' conditional Cooperation does not
require the Cooperation of all the other players, provided that for some
of the players this is required.

We note in passing that if(B„, B n , . . ., B„) is an equilibrium, it is also a
coordination equilibrium. The remarks in the section on Coordination
Equilibria in chapter 3 apply here also.

All these results still hold if a subset of the players use the
unconditionally Cooperative strategy C 00 ; only a change of detail in the
conditions to be met by the discount factors is necessary. Thus, strategy
vectors of the form (B„, . . ., B n ; C°°, . . ., C°°) can be equilibria. If every
player Cooperates on the condition that all other players Cooperated in

THE N-PERSON PRISONERS' DILEMMA SUPERGAME

the previous game, then if any player changes strategy to D x , all the
conditionally Cooperative players choose D in the second constituent
game, and hence in the third, and so on. It is easily checked that, if one of
the B n players defects, the condition for such a switch to yield a gain is the
same as inequality (4.1) but with 0(0) replaced by g(n c ), where n c is the
number of unconditional Cooperators. Call this new inequality (4.2). If
one of the C°° players defects to D 00 , g(0) is replaced by g(n c - 1 ). Since we
have assumed that g(v) > g{0) for any strictly positive value of v (i.e.,
Defection yields a greater payoff when some others Cooperate than it
does when nobody else Cooperates), we have g(n c ) > g(0) and
g{n c - 1) > 0(0). Thus the discount rate must now satisfy stricter
requirements than previously - so that, as is intuitively obvious, if (B m
. . ., B n ; C 00 , . . ., C 00 ) is stable against defection to D°°, then so also is (B m
. . ., B„). If we also assume, quite reasonably, that g(y) is strictly
increasing with v, then g(n c ) > g(n c - 1), and therefore if (B m . . ., B n ; C 00 ,
. . ., C") is stable against defections to D°° by a B„ player, it is stable
against such defections by a C x player, so that if (4.1) is satisfied when
0(0) is replaced by g(n c ), it is satisfied when #(0) is replaced by g(n c - 1).
As before, no strategy can do better than D 00 or B„ when all others are
playing B„ or C°° strategies (with n = N- 1 for all the B„ players). Thus,
strategy vectors of the form (B m . . ., B„ ; C =°, . . ., C 00 ), where n = N-lfor
all the players, are equilibria if and only if condition (4.2) is satisfied. If the
values of n vary amongst the players, the remarks made earlier about (B m
. . ., B„) apply here mutatis mutandis.

Subgroups of Cooperators

So far, we have merely generalized the basic result of the two-person
analysis. But an interesting new question arises in the JV-person case : is it
ever rational for a subset of the players to sustain Cooperation
throughout the supergame? Consider, then, strategy vectors of the form:

(B„, . . ., B„; C°°, . . ., C°°; D°°, . . ., D m ).

I will also write this as (BJC^/D* ) for short. Can such a strategy vector
be an equilibrium?

Let the number of players in these three groups be n B , nc and no, with
n B + n c + n D =N, and let n B + nc, the total number of Cooperators, be m.
If a strategy vector of this kind is ever to be an equilibrium, n B must not

THE N-PERSON PRISONERS' DILEMMA SUPERGAME 89

be zero, because a strategy vector in which there are only C 00 and D°° is
never an equilibrium. The following analysis covers as a special case
strategy vectors in which there are only B„ and D°° players (i.e., n c =0).

As before, the values of n are crucial. Consider first the case when
n < m — 1 for every player. All the first m players choose C in every
constituent game, for at each stage their Cooperation is conditional
upon at least n other players Cooperating, and since there are sufficient
Cooperators in the first game, so there are in the second, and therefore
also in the third, and so on. Now, since the Cooperation of each of the B„
players is conditional upon fewer than all the remaining Cooperators, if
any member of the B n and C 00 subgroups unilaterally defects to D x , the
remaining m — 1 conditional Cooperators continue to choose C in every
constituent game. The defector's payoff will therefore increase, for
g(m-l) > f(m-l), that is, a player is better off if he Defects while
everyone else Cooperates than if he Cooperates. This is true, of course,
regardless of the values of the players' discount factors and of the shapes
of the payoff functions /(v) and g(v). Thus, strategy vectors of the form
(B n /C cc /D*') are never equilibria if n < m — 1 for every B„ player, whether
or not the values of n vary between the players.

With strategy vectors of this form we must also consider the possibility
that n > m — 1. Suppose n > m — 1 for every B„ player. In this case, even
before any of the Cooperators defects to D°°, the 'compact' amongst the
B„ players collapses immediately. All of the first m players choose C in
the first constituent game; thereafter all conditional Cooperation
collapses, for the total number of Cooperators in the first game (m) is too
small. Thus, a B n or C°° player who defects to D"° will necessarily gain,
since his payoff in the first game is greater, while in every succeeding
game his payoffs are the same. Thus, strategy vectors of the form
(BJC™ ID™) are never equilibria if n > m — 1 for every player.

The point about these two cases - in which n < m — 1 for every B„
player or n > m — 1 for every B„ player - is that a defection to D°° by one
of the Cooperators makes no difference to the subsequent actions of any
of the other Cooperators. This is not so in the remaining cases, and we
shall now show that under certain conditions strategy vectors of the form
(BJC^/D^) can be equilibria.

The first of these cases is where n = m — 1 for all the B„ players, that is,
each B n player chooses C as long as all the other Cooperators,
conditional and unconditional, chose C in the preceding game. Before

THE N-PERSON PRISONERS' DILEMMA SUPERGAME

any player changes strategy, there are exactly enough players choosing C
in each game to maintain throughout the supergame the 'compact'
amongst the conditional Cooperators. The first m players (which we will
index i = 1, . . ., m) choose C in every constituent game and the remaining
N-m players (indexed = m + 1, m + 2, . . ., N) choose D in every game.

But if any one of the B„ or C°° players (/ = 1, say) changes strategy to
D 00 , then the 'compact' amongst the rest of these players collapses: after
choosing C in the first game, all these players choose D in the second
game, because in the first game the total number of other Cooperators
was m — 2; they therefore also choose D in the third game; and so on. The
defector, then, receives a greater payoff in the first game (in which he is
the lone Defector) than he did before changing strategy, but receives less
in every succeeding game. Whether his total discounted payoff from the
supergame is greater depends as usual on his discount rate and on the
constituent game payoffs. A B„ player who defects to D w does not gain if
and only if

a: a 2 i

f(m-l)- > g(m- l)a t + g(n c )z — 3

1 — a t l — a i

that is,

o(m-l)-/(m-l)

a,- ^ (4.3)

g(m-\)-g(n c )

(Condition (4. 1 ) is a special case of this : if there are no C 00 or D 00 players,
then n c = and m— 1 =N — 1, and (4.3) becomes (4.1)).

A change of strategy by a B„ player to C°°, or by a C°° player to B m
makes no difference, of course, to anyone's choices throughout the
supergame.

If one of the C*> players defects to Z)°°, then as before the cooperation
amongst the B„ players collapses after the first game; the number of
players choosing D in the first game is the same as before (when a B n
player switched to D°°) but there is one less C-chooser from the second
game onwards. So the condition for a C x player to make no gain from
defecting to D x is (4.3) but with n c - 1 rather than n c . Assuming that
g(n c ) > g(n c -l), this modified condition is satisfied if (4.3) is. So a
strategy vector of the form (BJC^/D 00 ) is an equilibrium only if condition
(4.3) is satisfied for all the B„ and C°° players.

As in the case of (B„, . . ., B„), if it does not pay a B„ player to defect to

THE N-PERSON PRISONERS' DILEMMA SUPERGAME

it will not pay him to defect to B' n (with n = m - 1 again). In fact, it is
easily verified that there is no other strategy which it would be better for a
B n player to defect to than D°°.

A defection by a C 00 player to B' n (with n = m- 1) results in the same
pattern of choices and yields the defecting player the same payoff as in
the case of a defection by a B n player to B'„.

And finally, it is clear that no change of strategy by a D x player would
yield him a gain. Whatever he did, it would have no effect on the choices
of any of the other players thoughout the supergame. If his new strategy
was such that in any constituent game he played C, he would therefore
obtain a lower payoff in that game than he did before switching -f(m)
instead of g(m) - while in any constituent game in which he played D his
payoff would be the same as before.

Thus, condition (4.3) for all i = l,2, . . ., m (i.e.,, for all the
Cooperators) is a necessary and sufficient condition for strategy vectors of
the form (BJC^/D™), where n = m — l for every B n player, to be an
equilibrium.

We have established, then, that strategy vectors of this form can never be
equilibria if n > m — 1 for every B„ player or n < m - 1 for every B„ player
(whether the values of n vary or not), but that if every n is m — 1 they are
equilibria if and only if (4.3) holds for all the Cooperators. There remain
all those cases where the values of n are distributed between n < m — 1,
n =m - 1 and n > m — 1, there being B„ players falling in at least two of
these categories.

If there are any B„ players with n = m-\, then their choices after the
first game will be affected (if only in the second game) by a switch to D°°
by one of the other B„ players. Such a switch can also affect the choices of
some of the B n players with n < m — 1, if there are any; this would depend
on the particular values of n in this group as well as on the numbers of
players using different values of n (the frequency distribution of n). Any
choices thus affected would of course change from C to D. The player
who switched may or may not gain, depending on the pattern of choices
resulting from his defection and depending (as usual) on his discount
rate.

If there are B„ players with n < m - 1 and some with n > m — 1 but
none with n=m- 1, then whether a unilateral switch to D°° by any B„
player affects other players' choices depends on the values of n in the

THE TV-PERSON PRISONERS' DILEMMA SUPERGAME

n < m — 1 group (the only group whose members could be affected) and
on the frequency distribution of n, and the defecting player may or may
not gain, depending on the pattern of choices resulting from his defection
and on his discount rate.

Similar observations can be made about defections to B'.

Establishing general conditions on the frequency distribution of n,
indicating which cases in these last two groups are equilibria (if also the
discount rates are not too great) is not very instructive. I have said
enough to show (and the reader can easily construct examples which
show) that some strategy vectors of this kind are equilibria, provided the
relevant discount rates are not too great. The conditions which the
discount rates would have to satisfy would depend on the frequency
distribution of n (which determines the pattern of choices) as well as on
the payoff functions / and g.

Chickens nesting in the Prisoners' Dilemma supergame

It is worth noting that (4. 1) does not imply and is not implied by (4.3). So
(B m . . ., B„) can be an equilibrium without a strategy vector of the form
(BJC a> /D'* > ) being one, and conversely, or both can be equilibria
simultaneously.

It both are equilibria, along with (D°°, . . ., D 00 ), and assuming for the
moment that there is no other equilibrium, which would be the
outcome? Any individual would prefer (B N - U . . .,B N - t ) to
(Bjv-i/C 00 /!) 00 ) if he was one of the Cooperators (conditional or
unconditional) in the second strategy vector; but whether a D°° player
has the same preference depends on whether f(N-l) > g(m), which is
not implied by our assumptions. This last inequality is increasingly less
likely to hold as m increases, that is, as the group of/) 00 players shrinks.
Again, every player prefers (B N - U . . ., B N - t ) to (D°°, . . ., Z)°°), but
whether any of the Cooperators prefers (B N - i /C x /D x ) to (Z)°°, . . ., £>°°)
depends on whether f(m-l) > g(0), which is not implied by our
assumptions (except when m = N, i.e., there are no D 00 players in the first
strategy vector).

If(B N - u . . .,B w _i) is the only equilibrium besides (£>°°,. . .,£>°°),then
it will be the outcome. But if strategy vectors of the form (fl m _ i/C 00 //) 00 )
are also equilibria, then even if these vectors are unanimously preferred

THE TV-PERSON PRISONERS' DILEMMA SUPERGAME

to (D°°, . . ., Z>°°) and even if 'there are no other preferred equilibria, it is far
from certain that one of them will be the outcome.

The problem lies in the multiplicity of equilibria. Given that the payoff
functions / and g are the same for all players, there could be as many as
O equilibria of the (B m _ ! /C 00 ) class, O being the number of subsets
of size m that can be drawn from the N players. 3 An individual is
indifferent between them all if he is going to Cooperate (conditionally or
unconditionally - it makes no difference) or if he is going to Defect (play
D°°). And each player would prefer to play B m _ x i/otherwise there would
be no Cooperation at all (i.e., if his Cooperation is critical to the success
of the 'compact' amongst the B m _ t players). But if one of the equilibria in
this class is going to be the outcome, then he would rather it was one in
which he played D°° than one in which he Cooperated. Every player has
this preference. If the public good is going to be provided by a subgroup,
he would rather it was a subgroup which did not include him.

Every player, then, has an incentive in this situation to pre-commit or
bind himself to non-Cooperation, in the expectation that others will have
to Cooperate and he will be a free rider on their efforts. The existence of
an incentive of this kind is precisely the defining characteristic of the N-
person Chicken game - according to the account given in chapter 2 at
least. As I noted in that earlier discussion, there is little that the game
theorist, or anyone operating within the framework of rational egoism
assumed here, can say definitively about the game of Chicken played
only once. (For some tentative suggestions, see the section in chapter 2
on 'Pre-commitment as a risky decision and the prospects for cooper-
ation in Chicken games'.)

It has been suggested that in this situation there would be a 'chaotic'
scramble in which each individual tries to ensure that he is not left
behind in a subgroup whose members are obliged (by their own
preferences) to provide the public good and that there is no reason to
expect a subgroup of the right size to emerge from this 'stampede'. 4 But
again it must be emphasized that there is no warrant (either in the model
studied here or in any alternative model offered by the proponents of this
view) for this conclusion; conclusive arguments of this kind could only
be made if pre-commitment behaviour were explicitly incorporated into
the model. In fact, a subgroup of conditional Cooperators might emerge
if enough players are sufficiently risk averse - but arguments of this sort
too must await a richer specification of the model. 5

THE N-PERSON PRISONERS' DILEMMA SUPERGAME

Other Cooperative equilibria

The generalized tit-for-tat strategy B„ is not of course the only strategy
which, if used by all the players, would sustain universal Cooperation
throughout the supergame. Consider, for example, the following JV-
person generalization of the strategy A kJ considered in the last chapter:
C is chosen in the first game ; it continues to be chosen as long as at least n
other players (N > n > 0) chose C in the previous game; if the number of
other Cooperators falls below n, then D is chosen for the next k games
(where k is a strictly positive integer); C is then chosen in the next game
no matter what the other players chose in the preceding game; it
continues to be chosen as long as at least n other players chose C in the
preceding game; when the number of other Cooperators next falls below
n, D is chosen for k + 1 games (where / is a non-negative integer); and so
on ; the number of games in which the other players are 'punished' for
Defection is increased by / each time ; and each time there is a return to C.
Call this strategy A kJ .

Note that when n=0, both A kJ and B„ degenerate into C 00 . 1 assume
henceforth that n > 0. In the limiting case when k -* oo, the 'punishment'
period lasts for ever. When n = N — 1, Cooperation is conditional upon
the Cooperation of all the other players. Thus, in the special case A N X ~ )
(the value of / is now irrelevant), the first Defection of just one other
player is enough to trigger 'eternal punishment', that is, it causes an A^~ }
player to Defect in every succeeding game. But although this strategy (or
its analogue for general non-cooperative games) is the only one that
some economists have paid attention to, it seems to me that it embodies
an 'implausible' threat, or at any rate a threat that is less plausible than
those of strategies A k<l when k is finite. 6

It is easily shown that strategy vectors in which every player uses A kyl
are never equilibria if n < N — l, but are sometimes equilibria when
n = N — 1 for every player. In the case when the values of k and / are the
same for every player, the necessary and sufficient conditions for this
strategy vector to be an equilibrium can be shown to be:

-^%<N-l)-*«» f0ra1 "'

where S, = S(k, I; a,-) is defined as in the last chapter. This condition is the
jV-person analogue of condition (3.3), which is the necessary and
sufficient condition for (A kJ , A kJ ) to be an equilibrium.

THE JV-PERSON PRISONERS' DILEMMA SUPERGAME

But let us derive this condition as a special case of the more general
equilibrium conditions for strategy vectors in which B„, C x and D x are
present as well as A\ ,.

As in the earlier case of strategy vectors of the form (BJC^/D*), if
n < m — 1 for all the conditional Cooperators or if all n > m — 1, then
unilateral defection to £)°° by any of the Cooperators (conditional or
unconditional) makes no difference to the choices made throughout the
supergame by any other player. Such a defection must therefore
necessarily pay: the defector's payoff in any constituent game either
increases from /(v) to g(v) or stays the same at g(v). Strategy vectors of
this sort can therefore never be equilibria if all n < m — 1 or if all
n > m — 1.

If n = m — 1, then there are exactly enough players choosing C in each
constituent game to maintain throughout the supergame the 'compact'
amongst the conditional Cooperators. If nobody changes strategy, the
first m players all choose C in every game, and the supergame payoff to
each of them is f(m- l)a ( /(l -a,), while the payoff to each of the D°°
players is g(m — l)a,/(l — a,). But if one of the Cooperators changes
strategy, switching to D in any constituent game, the 'compact' collapses
for the rest of the supergame. If one of the Cooperators defects to D°°,
then his payoff becomes

g(m-\) ai + g(n A + n c -d) (St-ad + ginc-S 1 ) -S^j

where n A and n c are the numbers of players using A\ , and C°°
respectively, m is the total number using conditionally or uncondition-
ally Cooperative strategies, S, = S(/c, /; a { ) is as defined earlier, and

_ fO if player i is using B„
[l if player / is using A kii or

and

if player i is using A n ktl or B„

1 if player is using C°°

Call this payoff P. Then a unilateral change of strategy to D x does not
yield one of the Cooperators a gain if and only if

f(m-l)j^->P
1 — a t

(4.4)

THE JV-PERSON PRISONERS' DILEMMA SUPERGAME

It is easily verified that, if it pays any player to change strategy at all, he
can do no better than switch to D°°. For, first, it does not pay a £> x player
to switch to any other strategy. And, second, if a change by any of the
Cooperators is to yield a gain it must be such that in some constituent
game (and therefore in the first) D is chosen; but this causes all the
(other) conditional Cooperators to Defect in the second game, and no
matter what the switcher does in this second game (even if C is chosen, as
would happen if the switch was to B' n for example) all the other
conditional Cooperators will Defect in the third game, and therefore in
all succeeding games; so in the second game (and in each of the following
games) the switcher can do no better than to play D. Hence, conditions
(4.4), for all i = l, 2, . . ., m, are necessary and sufficient for strategy
vectors of the form (A^/BJC^/D 00 ) to be equilibria.

When k, I and n vary amongst the players, an enormous variety of
patterns of choices is possible. In particular, there can easily arise a
pattern in which Cooperation amongst some of the B n players rises and
falls throughout the supergame (as a result of the periodical return to
Cooperation of the Alj players). It turns out that some strategy vectors
of this form are equilibria under certain conditions, which are stringent.
The details of the analysis are messy and will not be set out here. The
reader interested in an illustrative analysis will find one in Anarchy and
Cooperation (at pp. 57-60).

An example

To illustrate some of the equilibrium results obtained so far, let us briefly
reconsider Hardin's formulation of the collective action problem in
terms of a (one-shot) Prisoners' Dilemma, first introduced in chapter 1.
In that formulation, each of the N players is supposed to choose between
contributing a unit of the cost of providing a public good (strategy C) or
not contributing (strategy D). For each unit contributed, the total benefit
to the group is r and the benefit to each individual is assumed to be r/N
(the good having been tacitly assumed to be rival). Obviously we must
have r > 1. All players benefit from a contribution by any player.
The payoff functions f{v) and g(v) can be specified as follows:

THE JV-PERSON PRISONERS' DILEMMA SUPERGAME

This is a Prisoners' Dilemma if and only if: (i) D dominates C for each
player; that is, g(v) >f(v) for all v ^ 0, which is true if and only if r < N;
and (ii) the outcome (C, C, . . ., C) is preferred by every player to (£>, D,
. . ., D); that is, f(N- 1) > 0(0), which is true if and only if r > 1.

Thus 1 < r < N is a necessary and sufficient condition for the game to
be a Prisoners' Dilemma, with D being each player's rational choice.

Suppose that this condition is met, and consider now the supergame
consisting of an infinite number of iterations of this game. The results
just obtained for the JV-person Prisoners' Dilemma supergame can be
applied, for the functions / and g satisfy the three assumptions on which
those results depend. The first two assumptions are simply those which
guarantee that the constituent game is a Prisoners' Dilemma; the third is
that g(v) > g(0) for all v > 0, which is satisfied here since r > 1.

Consider first the strategy vector in which everyone plays B„. This was
shown to be an equilibrium if and only if n =N — 1 for every player (i.e.,
everyone Cooperates if and only if everyone else does) and condition
(4.1) holds for each player, that is:

g(N-l)-f(N-l)

t xT n ?ST" foralU

g(N-l)-g(0)

Substituting for / and g, this becomes

N-r

a t >

(N-l)r

From this, the required minimum discount factor can be calculated for
any values of N and r. Note that if there is virtually total discounting
(i.e., at -* 0), this equilibrium condition is satisfied for r > N, that is, N
units of public good benefit must be produced out of each unit of contri-
butions. If there is virtually no discounting (i.e., a, -»l), then the
equilibrium condition is satisfied for r > 1, that is, for any value of r. This
is just as we would expect. If very great public good benefit can be
produced out of each unit contributed, it will pay conditional Cooper-
ators to produce the public good even if their discount rates are very
great, and conversely if discounting is negligible, conditional Cooper-
ation will pay even if a unit of contributions only just produces more
than a unit of public good benefit.

Now consider strategy vectors in which the four strategies Al, h B m C°°
and D°° all appear, with n = m — 1 for all the conditional Cooperators.

THE Af-PERSON PRISONERS' DILEMMA SUPERGAME

We saw earlier that a necessary and sufficient condition for a strategy
vector of this form to be an equilibrium is the inequality (4.4).
Substituting for / and g, condition (4.4) reduces to:

"< " S ') + - (£) * 7 fe)- S ' f0 ' P ' ayerS USi " 8 A '"

^ ?L ( a ' | _ a f or players using B„
r \l-aj

2s — ( a ' ) — for players using C 00

r \\-aJ l-at

These inequalities being independent of m, equilibrium is dependent
only on the number of players using conditional strategies (subject of
course to n A + n B < m < N). Notice, too, that since

a, < S(k, I; a,) <

1-a,-

for non-zero finite k, the smallest values of n A (given n B ) and n B (given n A )
sufficient for defection to be irrational are greater for players using B„ (or
4£,,) than they are for those using A n Kl (for any non-zero finite k), and
greater for the latter than for those using C°°. This is as one would expect.

To illustrate more concretely, assume that in all the strategies of the
A n u type, fc = 3 and /=1, and a,=0.9, N = 100 and r = 5. With these
values of k and /, we have

S(k, I; a i ) = a i + a 5 i + a 1 i °+a i i 6 +. . .

and with a f =0.9 this is found to be approximately 2.17. Substituting for
S h a b N and r, the three inequalities above become:

n A + 1.186h b ^ 26.04 for players using Al A
^ 26.22 for players using B„
25.04 for players using C 00

Thus the second of these inequalities is necessary and sufficient for the
strategy vector to be an equilibrium.

We see that strategy vectors of this type will not be equilibria if the
total number of conditional Cooperators is too small. The general point
is that, if there are too few conditional Cooperators, then it pays any of
the players not using £>°° to change to that strategy, because such

THE iV-PERSON PRISONERS' DILEMMA SUPERGAME 99

defection leads to the defection of only a few other players and therefore
(in view of the assumptions about the payoff functions / and g) to only a
small loss of benefits, which is more than compensated by not having to
pay the cost of Cooperating.

Alternation between blocs of conditional cooperators

In the analysis of the two-person Prisoners' Dilemma supergame we saw
that the strategy vector (B, B'), which produces alternation between
(C,D) and (D,C) thoughout the supergame, was of special interest.
There were good reasons to believe that it could be an equilibrium and at
the same time preferred by both players to the strategy vectors pro-
ducing mutual Cooperation throughout the supergame. Let us consider
an analogous N-person strategy vector, in which some players use B and
the rest use B'.

A typical B„ player will be labelled i and a typical B'„ player will be
labelled j. Let the number of players using B n be n B as before. Then there
are N — n B players using B' n . It is still assumed that the same value of n is
used by all N players. The addition of players using C°° and D°° would
make no essential difference (as the earlier analysis of (A™ JBJC^/D 00 )-
type strategies should make clear), although of course the actual payoffs
and inequalities obtained below would be different.

Three cases have to be considered separately: (1) n < n B : (2) n > n B ;
and (3) n = n B .

(1) n < n B

The players using B„ choose C in the first game, while those using B' n
choose D. But there are enough players in the B n group (since n B — 1 ^ n)
to guarantee their own continued Cooperation in the second game and
also to cause the B' n players to Cooperate in that game. From the second
game onwards, all players Cooperate. We can see immediately that these
cases can never be equilibria, because (whenever n < n B ) a change by one
of the B'„ players to D 00 has no effect on the choices of the other players:
they still Cooperate from the second game onwards. Hence such a
change yields a greater payoff for the defector, for while his payoff in the
first game is unchanged, his payoff in each succeeding game is increased
from/(AT-l) to g(N-l).

100

THE N-PERSON PRISONERS' DILEMMA SUPERGAME

(2) n > n B

Here all players Defect from the second game onwards. If a player using
B„ changes his strategy to D°°, his payoff is thereby increased, for in the
first game he receives g(n B — 1 ) whereas before he received f{n B — 1), and
in all subsequent games his payoff is unchanged. Thus, strategy vectors
in this category are never equilibria.

(3) n = n B

The B n players choose C in the first game but Defect in the second. The B' n
players choose D in the first game but Cooperate in the second. What
happens in the rest of the game depends on the relation between n and
the number of players using B' n . This is examined in the following
subcases.

(3.1) n<N-n B (so that n B < jN)

Then there are enough B' n players to bring back the B„ players to
Cooperation in the third game and to maintain the Cooperation of
the B„ players themselves. All N players choose C from the third
game onwards.

(3.1.1) n<N-n B -l

If one of the Bj, players changes his strategy to then the choices
of the other players are unaffected. He therefore gains by such a
change, for although his payoff in the first game is unchanged, his
payoff in the second and every succeeding game increases. Thus,
strategy vectors in this category are never equilibria.

(3. 1.2) n = N — n B — 1 (thus n B = \(N - 1 ), so that N must be odd).

In this case a change of strategy by one of the B„ players to D 00 or
to B' n does not leave the other players' choices unaffected. It results
in all N players Defecting from the second game onwards. Such
a change does not yield the defector a greater payoff if and only if

g(n B -l) ai +g(0}— '— ^
l-fl f

a 3

J[n B - Vat + giN-ngrt+flN - (4.5)

1-a,

THE JV-PERSON PRISONERS' DILEMMA SUPERGAME

101

A change of strategy by one of the B' n players to Z) 00 results in the B„
players alternating between C and D throughout the supergame
beginning with C in the third game, and the B'„ players alternating
similarly but beginning with D in the third game. This does not
yield a greater payoff for the defector if and only if (after
simplifying)

g(n B ) ^f(n B )(l-aj)+f(N-l)aj (4.6)

If one of the B,; players changes his strategy to B„. then this results
in the B„ players choosing C in the second game but no other
choices are changed. Thus, such a change does not yield a greater
payoff if and only if

g(n B )>f(n B )(l-aj)+f(N-l)aj

but this is the reverse of inequality (4.6) above.

It can be shown (using reasoning of the kind we've followed
before) that a player switching unilaterally from B n or B,| cannot
do better than to switch to the better of D°° and B' n or B„. Thus,
necessary and sufficient conditions for strategy vectors in this
category are the restrictive condition

g(n B )=f(n B )(l-aj)+f(N-l) aj (4.7)

(for each of the B' n players) together with inequality (4.5) for each of
the B n players. If f(N - 1 ) > f(n B ), the equality (4.7) can be written
in the form

gin B )-f(n B )
° J f(N-l)-f(n B )

Clearly, then, strategy vectors of this form (which result in
universal Cooperation from the third game onwards) are most
unlikely to be equilibria, since this requires that this same equality
be exactly satisfied by the discount factors of every one of the B' n
players, as well as inequality (4.5) holding for all the B„ players.

n > N-n B (so that n B > \N)

In this case, before any player changes strategy, all N players
Defect from the third game onwards. If one of the B' n players
changes his strategy to Z)°°, no other player's choices are affected,

THE JV-PERSON PRISONERS' DILEMMA SUPERG AME

and the defector's payoff changes only in the second game, where it
increases. Thus, strategy vectors in this category are never
equilibria.

3) n = N-n B (thus n B =\N, so that JV must be even)

In this case, before any player changes strategy, each player's
choices alternate between C and D throughout the supergame, the B„
players beginning with C and the B' n players beginning with D.

If one of the B n players changes his strategy to D°° or B'„, then all
JV players Defect from the second game onward. The defector's
payoff is not increased if and only if

2 2 2

f(n B ~ 1) j~ + g(N-n B ) > g(n B - l)a i + g(0)

l — a i 1 — a~- 1 — a,

If one of the B' n players (j) changes his strategy to B n , then all JV
players Cooperate from the third game onwards, and /s payoff is
not increased if and only if

2 2

9(»b) +AN-n B -l) > f(n B )aj +f(N - 1 ) -^L-

l-a) l-a) l-aj

If one of the B' n players (J) changes his strategy to Z)°°, then all JV
players Defect from the second game onwards, and fs payoff is not
increased if and only if

g(n B )-^ +f{N - n B - 1 ) >

a 3 -

g (n B )aj + g(N-n B -l)ePj + g(0)—l-

1-a,

The three inequalities given above (the first to hold for each of
the B„ players, the second and third for the B' n players) are
necessary and sufficient conditions for the strategy vectors in this
category to be equilibria. Examples can be found to show that
these three conditions can be satisfied simultaneously. (For
example, when the payoff functions / and g take the form of the
example discussed in the last section, there are permissible values

THE N-PERSON PRISONERS' DILEMMA SUPERGAME

103

of JV (and hence n B ) and r and of the players' discount factors for
which the conditions all hold.)

This completes the analysis of strategy vectors in which some players use
B„ and the rest use B'„. It shows that there are two sets of circumstances in
which these strategy vectors can be equilibria : when n = n B and JV — n B is
either n B + 1 or n B . In the first case, JV is odd and the number of B'„ players
is just one more than the number of B n players. In this case, all the players
Cooperate from the third game onwards. But in addition to the
requirements on the relations between n, n B and N — n B , the discount
rates for the B' n players must satisfy very exacting conditions. So
equilibrium is very unlikely in this case. In the second case, JV is even and
there are exactly as many B' n players as there are B„ players. These two
equal blocs alternate between C and D : in one constituent game all the
members of the B n bloc play C while all those in the B' n bloc play D; in the
next game the opposite happens; and so on. If the provision of a public
good were at stake here, this alternation could consist of the two blocs
'taking it in turns' to provide the public good, that is, the members of the
two blocs contributing in alternate time periods. This is a perfectly
plausible set-up, at least for small groups.

It is easily shown that when (B m . . .,B n ;B' m . . ., B' n ) is an equilibrium, it
is preferred to (Z) x , . . ., D°°) under some conditions (which of course
relate discount rates to constituent game payoffs). It is possible that (B„,
. . ., B„) is simultaneously an equilibrium, but this is exceedingly unlikely,
for then (as in the two-person case) the discount rates would have to
satisfy a set of exact equalities. But even assuming that strategy vectors of
the form (B„, . . ., B„; B'„, . . ., B^) are Pareto-preferred to both (Z) 00 , . . .,
D 00 ) and any strategy vectors, such as (B„, . . ., B„), which give rise to
universal Cooperation throughout the supergame, there still remains a
problem of the kind we encountered in the two-person case : each player
would prefer an equilibrium in which he was in the B' n bloc (and
therefore Defected in the first game) to one in which he was in the B„ bloc
(and therefore Cooperated in the first game). So even if all players
eliminated from consideration all strategies except B„ and B' m they would
still face the 'coordination' problem of avoiding (B n , . . ., B„) and (B' m . . .,
B^) as well as the problem of their conflicting preferences over the
different alternation equilibria. Again, within the framework we are
using here, there is no resolution of this problem.

104 THE JV-PERSON PRISONERS' DILEMMA SUPERGAME

Summary and discussion of results

The account given above of the N-person Prisoners' Dilemma super-
game is far from complete but suffices, I think, to indicate the broad
conclusions which a more comprehensive analysis would yield. I have
carried the discussion far enough to show how Cooperation can arise in
the Prisoners' Dilemma supergame, no matter how many players there
are. This is the main point I set out to establish and it would not be
undermined by a more general analysis.

We have seen that if Cooperation is to occur at all, then at least some
of the players must be conditional Cooperators. More specifically, we
saw in the first place that Cooperation by every player throughout the
supergame, sustained by the use of the tit-for-tat strategy B„ by each
player, is an equilibrium when each player's Cooperation is conditional
on that of all the other players and his discount rate is not greater than a
certain function of the constituent game payoffs (condition 4.1).

Secondly, however, we saw that even when some of the players insist on
unconditional Defection throughout the supergame, Cooperation may still
be rational for the rest - provided that there are some players who
Cooperate conditionally on the Cooperation of all the other Cooper-
ators, both conditional and unconditional, and that all the Cooperators'
discount rates are not too great. But even when strategies of this sort are
equilibria, it is difficult to say confidently what the outcome will be,
because there are many such equilibria and each player prefers an
outcome in which he is an unconditional Defector to one in which he is a
Cooperator. This gives rise to a game of Chicken, whose outcome, as we
saw in chapter 2, is ill-determined.

The third result of interest is that a pattern in which two equal blocs of
players using strategies B n and B'„, which results in them 'taking it in
turns' to Cooperate throughout the supergame, can be an equilibrium
under certain conditions. The conditions, however, are rather exacting.

It plainly cannot be concluded from these results, and those of the last
chapter, that the 'dilemma' in the Prisoners' Dilemma is resolved upon
the introduction of time and the interdependence of choices over time :
that people who would not Cooperate in the one-shot game will do so in
the supergame. Nevertheless, it has been shown that under certain
conditions the Cooperation of some or all of the players could emerge in
the supergame no matter how many players there are. The question arises,
whether these conditions are likely to be met in practice.

THE N-PERSON PRISONERS' DILEMMA SUPERGAME

105

Speaking informally (which is all we can do here), it is pretty clear that
Cooperation amongst a relatively large number of players is 'less likely'
to occur than Cooperation amongst a small number. For a start, the
more players there are, the greater is the number of conditions that have
to be satisfied - the conditions specifying that the right kinds of
conditionally Cooperative strategies are present and those specifying the
inequalities that all the Cooperators' discount rates must satisfy. But the
main reason for this new 'size' effect is that Cooperation can be sustained
only if conditional Cooperators are present and conditional Cooper-
ators must be able to monitor the behaviour of others. Clearly, such
monitoring becomes increasingly difficult as the size of the group
increases. (It is true that, in the supergame model we have analysed, a
conditional Cooperator needs only to know that at least a certain
number of others Cooperated in the preceding game; he does not need to
know which other players Cooperated. This reduces but does not obviate
the need for monitoring.) The required monitoring is more likely to be
possible in a very small group, especially one with an unchanging or very
slowly changing membership, or in a community. 7 On the account of
'community' I gave in chapter 1, communities tend to be small, but not
necessarily very small. Monitoring and hence conditional Cooperation
are made possible in communities in part by their relatively small size
but also by the quality of relations between their members. As I argued in
chapter 1, however, this is not always enough to guarantee the success of
conditional Cooperation in the provision of public goods. In groups of
intermediate size, including most communities, conditional Cooper-
ation needs to be facilitated and supported by the deployment of
(positive and negative) sanctions additional to those sanctions which
may be thought to be embedded in conditional Cooperation itself. The
additional sanctions need not, however, be centralized. 8

A more realistic model

My aim in this book is to criticize what I believe is the strongest and most
popular argument for the desirability of the state. I make no attempt to
provide a positive theory of anarchy or even an indication of how people
might best provide themselves with public goods. I am therefore not
concerned with developing a detailed dynamic model of public goods
provision. I have merely tried to show that, even if we accept the

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THE N-PERSON PRISONERS' DILEMMA SUPERG AME

pessimistic assumption (an assumption unfavourable to the case I am
making out) that individual preferences have the structure of a
Prisoners' Dilemma at any point in time, mutual Cooperation over time
may nevertheless take place. However, the scope of the application of
this part of my critical argument would be increased if the broad
conclusions of the analysis in this chapter and the last could be shown to
apply to a more detailed, more realistic model of the dynamic process of
public goods provision than the Prisoners' Dilemma supergame model
(with or without some form of altruism) considered in those two
chapters. In this section, I want briefly to indicate, then, some ways in
which the supergame model might be extended or revised so as to
provide a more realistic picture of public goods interaction. I shall not
give any analysis of the alternative models or even specify them fully; to
do so would require another book.

In the Prisoners' Dilemma supergame considered above, it is assumed
that the time taken for a player to change from one strategy to another is
zero. There is no time-lag between the decision to change and the actual
change. Each player is perfectly 'flexible' in this respect, and therefore, in
particular, his strategy choice in any ordinary game can be contingent
upon the other player's choice in the immediately preceding game. In the
real world, this sort of flexibility may exist in decisions such as those to
commit or refrain from committing anti-social acts (a decision can be
translated immediately into 'action', with instantaneous production of
the public good or bad, social order or disorder), but with respect to
many kinds of public goods, flexibility is less than perfect. People must
be trained, equipment obtained, public works constructed, and so on,
before there is any benefit to anybody.

Intuitively, it would seem that the presence of time-lags of this sort
would tend to increase the 'likelihood' of non-Cooperation in the
Prisoners' Dilemma supergame; for a player (A) contemplating Defec-
tion from mutual Cooperation knows that it will take the other player
(B) several time periods to change his strategy after observing A's
Defection, and can expect the unilateral Defection payoff during this
interval; this will offer A a greater compensation for the mutual
Defection payoffs which will be his lot after Fs eventual Defection than it
would in the model analyzed here, where A can 'exploit' B (receive the
unilateral Defection payoff) for only one ordinary game. Thus, the
conditions for Cooperation to be rational are likely to require a

THE JV-PERSON PRISONERS' DILEMMA SUPERGAME 107

progressively smaller discount rate as this flexibility decreases (that is, as
the time-lag between decision and effect increases). 9

A second kind of inflexibility which may be present in individual
decisions in public goods interaction is the limitation on the frequency
with which strategy choices may be changed. If every strategy of every
individual could be changed equally regularly, no matter how in-
frequently, the supergame models studied here would not have to be
modified, since the durations of the time periods specified here are
essentially arbitrary. However, if only some strategy changes could be
made at any time (in any time period) while others could be changed less
frequently (for example, a decision to contribute to the provision of a
public good for the first time could perhaps be made at any time, but a
decision to cease contributing - to switch from Cooperation to
Defection - might be possible less often, because, for example, resources
have been committed), then the model would have to be modified. Again,
I suspect that the result of the modification would be to render more
restrictive the conditions for mutual Cooperation to be the outcome of
the supergame.

Perhaps the most important shortcoming of the Prisoners' Dilemma
supergame as a model of the process of public goods provision is that it
takes place in a static environment : the supergame consists of iterations
of the same ordinary game. In some of the public goods problems of
interest here, a more realistic description of reality would require a
changing payoff matrix, possibly a changing set of available strategies,
and even a changing set of players. These changes, especially the first,
might be the result of influences external to the game or of the history of
strategy choices of the players themselves. Where, for example, a
'common' (of the kind discussed in chapter 1) is being exploited, the
payoffs might decrease steadily as more and more non-Cooperative
('exploitative') choices are made over time; they might radically change
quite suddenly with the ecological collapse of the 'common' following a
long succession of non-Cooperative choices; and, for the same reasons,
the set of available strategies might become restricted and some of the
players might be obliged to withdraw from the game.

The possibilities here are very numerous, and it is impossible to make
any general statements about the effects of extensions of this sort on the
conclusions of the analysis in this chapter and the last. These effects
would very much depend on the particular manner in which the game

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THE TV-PERSON PRISONERS' DILEMMA SUPERGAME

changed over time. Perhaps the most important class of changes is the
one suggested above: all payoffs decline as a result of non-Cooperative
choices (the greater the number of players Defecting, the greater the
decline), and all payoffs increase, or at least do not decrease, as a result of
Cooperative choices. It seems very probable that an analysis of this sort
of dynamic game would show that mutual Cooperation is a more likely
outcome than in the 'static' supergame studied here; that is to say, a
lower discount rate than in the present model would suffice to make
Cooperation rational, for the gains from unilateral Defection from the
mutual Cooperation position (assuming that conditional strategies are
being used so that this unilateral Defection would cause other players to
Defect also) would clearly be smaller, other things being equal.

5. Altruism and superiority

Up to this point, the assumption has been made that every individual is a
pure egoist, concerned only to maximize his own payoff. I do in fact
believe that this is a good approximation to actual behaviour in very
many situations where collective action is a possibility. If it were not,
there would not be the innumerable unsolved collective action problems
that we can see around us in the world and there would not be as much
need as there obviously is for action by states, political entrepreneurs,
organizations and other 'external' agents in the solution of collective
action problems for, as we shall see, most collective action problems
would not arise or would be solved 'spontaneously' if there was enough
altruism. (The division of solutions to collective action problems into
'external' and 'internal' or 'spontaneous' was made in chapter 1.)

Olson himself went astray here, arguing that if it was not rational for a
pure egoist to contribute to the provision of a public good then it would
not be rational even for a pure altruist to do so, since his contribution to
the welfare of others 'would not be perceptible'. A rational altruist would
still want to allocate his resources where they had some effect (on
others). 1 This is nonsense. In the first place, although a contribution (e.g.,
work effort, or refraining from dumping rubbish) may not in fact be
noticed by others, or if noticed may not have any effect on them in the
sense of causing them to contribute, it is not literally 'imperceptible', nor
is its effect 'infinitesimal'. Otherwise it would not be possible for a (finite!)
number of people, however numerous, collectively to provide any of the
public good - or any 'noticeable' amount of it at any rate. 2 Secondly, no
inference can in any case be made from what is rational for a pure egoist
to what is rational for a pure altruist or for someone who combines
altruism with egoism. In a two-person situation, for example, my benefit
from the public good provided out of my contribution might just be

109

110

ALTRUISM AND SUPERIORITY

outweighed by the cost of my contribution, but your benefit from my
contribution costs you nothing. If, for example, an egoist's preferences
(his own benefits less his own costs) were those of a 2 x 2 Prisoners'
Dilemma, then he would choose D, but a pure altruist - out to maximize
the other's payoff - would choose C. And thirdly, the altruist, in
contributing to a non-excludable good, augments the utilities of
everyone else in the group, and it is the sum, or some other sort of
aggregate, of all of these utilities which (together with his own payoff if he
combines altruism with egoism) he compares with his costs. In a large
group, this might very well cause him to contribute.

There are two other ways in which altruism can make a difference to
the provision of public goods and the solution of collective action
problems generally. If some individuals contribute because they are
sufficiently altruistic (or for that matter for non-instrumental reasons,
such as 'self-expression'), then they might thereby provide a 'starter'
around which conditional cooperation by others who are rational
egoists can develop. If, for example, the conditions established in chapter
4 for conditional cooperation amongst all or some players in a Prisoners'
Dilemma supergame are not met, then a small number of players who
would not otherwise have cooperated might be induced to do so because
their cooperation is conditional on at least n others cooperating and the
altruistic cooperators constitute a subgroup of at least that size. This
might trigger further cooperation on the part of those whose cooper-
ation is conditional on the cooperation of more players than the group of
altruists contains. And so on. The second way, or group of ways, in which
altruism can make a difference is that the core of altruists may be able to
provide the wherewithal to bring about an 'external' solution of the
collective action problem amongst the remainder. For example, the
altruists' contributions might 'finance' a political entrepreneur, who goes
to work on the non-altruists. Or the altruists might shame the others into
cooperation or bring other informal social sanctions to bear on them.

In asserting that innumerable, important collective action problems
go unsolved or need external help for their solution because people are
insufficiently altruistic, I do not mean to say that people are never
altruistic. There clearly is a great deal of altruistic behaviour. But what I
would argue is that departures from egoism (and more radical depar-
tures from 'thin' rationality, for example those involving expressive

ALTRUISM AND SUPERIORITY

111

rather than instrumental motivation) are, as it were, luxuries. They are
less likely to occur where the courses of action available to an individual
are limited and the incentives affecting him are well-defined, clearly
apparent and substantial, and above all where the individual's choice
situation combined with his benefits and costs are such that a lot (in his
eyes) turns on his choice. In such situations of relative scarcity and
constraint, substantial altruism is unlikely. 3

In this chapter I shall consider very briefly what difference the
introduction of altruism makes to collective action problems, or more
specifically to behaviour in Prisoners' Dilemma and Chicken games. Of
special interest will be games in which each player is concerned with the
difference between his own payoff and the payoffs of other players -
which I will call his eminence, after Hobbes, or his superiority - as well as
with his own payoff per se. The pursuit of superiority involves 'negative
altruism' (my utility increases as your payoff decreases) as well as egoism
and is what Hobbes took people to be most concerned with, as we will
see in the next chapter. It is also, I fear, of particular importance in
relations between states.

The altruism considered here takes a very simple form. 4 Each player is
assumed to maximize an additive function of his own payoff and the
other players' payoffs. Just as long as the weight attached to any other
player's payoff is non-zero, I shall say that the individual acts altruisti-
cally. I shall also consider briefly what I will call 'sophisticated altruism',
where a player's utility depends upon the other players' utilities (which
might incorporate some form of altruism) as well as their payoffs.

I emphasize that 'altruism' here is confined to a regard for other
persons' payoffs and utilities. A player's utility is not dependent upon
anything else to do with the other players - for example their strategy
choices per se. (A conformist, for example, might value doing what others
do regardless of their payoffs.)

In what follows, some simple ideas and results are introduced which
will be of use in the two final chapters of the book. I have, however, gone
a little further than is necessary in this respect. Nevertheless, only a brief
introduction to this subject is given. It is mainly confined to the two-
person game. In the JV-person game, the different forms which altruism
can take are almost limitless in number ; of these I shall consider only one
which is of special interest.

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ALTRUISM AND SUPERIORITY

Altruism in two-person games

Consider a 2 x 2 one-shot Prisoners' Dilemma game in which the two
players' payoffs are pi and p 2 . The general form of the assumption of
altruistic behaviour is that each player i maximizes the following utility
function :

". = w.(Pi, Pi)

More specifically, I assume that each player maximizes a weighted sum of
Pi and p 2 :

u,- = aiPi + PiPj (i, 7 = 1, 2; i" #j)

Clearly, whereas it has sufficed up till now to assume only that each
player's payoff scale is unique up to a linear transformation, now we
must assume also that each player is able to compare other players'
payoffs with his own. He must be able to place the origins and units of the
payoff scales of the other players in a one-to-one correspondence with
those of his own payoff scale. (To make such comparisons, a player must
of course know the other players' payoffs. This is already a strong
assumption, at least for some applications; but little could be said about
altruism with it.)

When ft = (and a, # 0), player i is called a pure egoist. In this case the
analysis is the same for all positive values of a,. If a, > 0, then, I choose
a, = 1. This is the case we have considered already (in the last chapter).
Similarly, for a, < 0, I set a, = - 1.

When a, = (and ft # 0), player / is called a pure altruist. When ft > 0, ;
the particular value of ft is irrelevant, so I choose ft = 1. Similarly, for
ft < 0, I choose ft = -1.

These are polar cases. In between lie those cases in which egoism and
altruism are present together. It is not assumed that an individual can be
altruistic only at the expense of being egoistic: a, and ft need not, for
example, be complements or inverses.

When ft > 0, player i"s altruism is said to be positive. When ft < 0, it is
called negative.

I am aware that 'altruism' is popularly limited to what I call 'positive
altruism', but it is convenient here to use the more literal, root meaning.
In particular contexts, 'positive altruism' and 'negative altruism' might
alternatively be called 'benevolence' and 'malevolence' (or various other
pairs of words). These expressions would, however, be quite inap-

ALTRUISM AND SUPERIORITY

113

propriate in other contexts, and I therefore use the more neutral terms.

Sometimes, 'egoism' is limited to the case when oc f > 0, but again it is
convenient here to use the term more literally.

When a; < 0, the player's behaviour might in various contexts be
called 'ascetic', 'anticompetitive', 'masochistic', and so on.

In what follows (except for the section on sophisticated altruism), I
will confine the discussion to symmetric games, in which the values of a,
and ft do not differ between the players.

Suppose that the payoff matrix is :

X, X

y, z

w, w

Then if each player's utility is a weighted sum of the two players' payoffs,
the resulting utility matrix is :

(ai+fti)x, (a 2 +ft.)x

a 1 z + fty, ix 2 y + p 2 z

a^ + ftz, ct 2 z + p 2 y

(«i+ftiK (a 2 +ft>)w

For emphasis, I shall speak of the resulting game as the transformed
game.

Suppose that the untransformed game is a Prisoners' Dilemma, that is,
suppose that y > x> w> z. What sort of game will the transformed
game be?

In the transformed game whose utility matrix is shown above, D
dominates C for player i if and only if

a,y + ftz > (oti + ft)*

and

(a, + ft)w > a.z + fty

that is,

«« > —A (5.1)

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ALTRUISM AND SUPERIORITY

and

y — W „

«<>- Pi (5.2)

w — z

Outcome (C, C) is (strictly) preferred to (D, D) by player i if and only if

(a i +p i )x> {<*.i+Pi)w

that is,

a, + ft>0 (5.3)

TTius, t/ie transformed game is a Prisoners' Dilemma, with (D, D), the only
equilibrium, being Pareto-inferior to (C, C), if and only if (5.1), (5.2) and
(5.3) are true for both players.

We see that for given values of x, y, w and z, the inequalities (5.1) and
(5.2) are 'more readily' satisfied as the ratio a,//?, increases, that is, as
player fs 'egoism' increases relative to his 'altruism'.

I examine now some cases of special interest - some more so than
others.

( 1 ) otj i ^ 0, ft :> (i = 1, 2) : each player's utility increases as his own payoff
increases and as the other player's payoff increases.

(1.1) Egoism and positive altruism (a, > 0, ft > 0, i= 1, 2): each player
cares about both his own and the other player's payoff; his utility
increases with increases in either of them, other things being equal. For
given values of x, y, w and z, a transformed Prisoners' Dilemma may or
not be a Prisoners' Dilemma. Inequality (5. 3 ) is always satisfied, but (5. 1 )
and (5.2) may not be. In fact (as the reader can verify) a Prisoners'
Dilemma can be transformed into a game of Chicken, an Assurance
game, or even a game in which C dominates D for both players. In the last
two eventualities, mutual Cooperation is assured and, as I argued earlier,
it is more likely in a Chicken game than in a Prisoners' Dilemma. So, as
one would expect, a measure of positive altruism may improve the
prospects for Cooperation. If there is only positive altruism, then, as
paragraph (1.2) will show, mutual Cooperation is necessarily the
outcome.

We saw in chapter 2 that individual preferences in many important
collective action problems are more likely to be those of a Chicken game

ALTRUISM AND SUPERIORITY

115

than those of a Prisoners' Dilemma. It is easily shown that positive
altruism can transform a Chicken game into one in which C dominates D
for each player; but (no matter what kind or degree of egoism it is
combined with) it cannot transform any Chicken game into a Prisoners'
Dilemma. Only positive egoism and negative altruism can do that.

(1.2) Pure positive altruism (oc,=0, ft > 1, i= 1, 2): each player desires
only to maximize the other player's payoff (or some proportion of it - it
makes no difference). Starting from a Prisoners' Dilemma or a Chicken
game, pure positive altruism produces a game in which C dominates D
for each player. So (C, C) is the outcome. Furthermore, it is preferred by
both players to (D, D). But although (C, C) is the only equilibrium, each
player's first preference is for the other player to Defect while he
Cooperates.

(2) a; ^ 0, ft < (i' = l, 2): each player's utility increases as his own
payoff increases and as the other player's payoff decreases.

(2.1) Pure negative altruism (a f =0, ft < 0, i- 1, 2): each player desires
only to minimize the other player's payoff. Starting from a Prisoners'
Dilemma or a Chicken game, pure negative altruism produces a game in
which D is each player's dominant strategy, and (D, D), which must be
the outcome, is Pareto-optimal.

(2.2) Egoism and negative altruism (a, > 0, ft < 0, i = 1,2). Starting with a
Prisoners' Dilemma, the addition of negative altruism to each
individual's egoism produces a game in which D dominates C for each
player (the inequalities (5.1) and (5.2) are always satisfied if the
untransformed game is a Prisoners' Dilemma). So (D, D) is the outcome.
But the transformed game is not necessarily a Prisoners' Dilemma, since
inequality (5.3) need not hold. A Chicken game, on the other hand, may
be transformed into a game with D dominant (with or without (C, C)
preferred to (D, D)) or into another Chicken game.

The following special case of egoism and negative altruism is
important.

(2.2.1) Games of Difference. These are perhaps the most interesting of the
transformed games. They are relevant to the account of Hobbes's

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ALTRUISM AND SUPERIORITY

political theory given in the next chapter. Suppose that each player's
utility increases both with his own payoff and with the difference between
his and the other player's payoff. He wants to increase his own payoff but
he also wants to increase his superiority over the other player, other
things being equal. Specifically, suppose that each player seeks to
maximize a convex combination of his own payoff and the excess of his
own payoff over the other player's. That is,

= KPi + (1 - Aj) (p, - pj), i = 1, 2, i #;

where s$ A, < 1 (i = l, 2). (When A, = l, we have the original un-
transformed game.) A f /(1 - A,), then, is the weight player i attaches to his
payoff relative to the excess of his own payoff over the other player's
payoff.

This expression for u, can be obtained from the general form,
Ui = «iPi+PiPj, by setting

*t = \,h = }*-\ (i = l, 2)

The transformed game, which I call a Game of Difference, is:

z-(i-Aib>, y-(i-A 2 )z

y-(l-A 1 )z,z-(l-A 2 )y

A x w, A 2 w

If we start with a Prisoners' Dilemma game, then in the transformed
game, as we have already noted in paragraph (2.2), D dominates C
whenever a f > and < (for /= 1, 2), which is the case here.

Thus (D, D) is the only equilibrium. But (C, C) is preferred to (D, D) by
player i, if and only if A, > 0. (When A f = he is indifferent between these
two outcomes.)

Thus, if the untransformed game is a Prisoners' Dilemma, the Game of
Difference is always a Prisoners' Dilemma just as long as (i=l, 2).

We shall see below that a similar result holds for an AT-person
generalization of this Game of Difference.

If we call a player's gain from defecting unilaterally from mutual
Cooperation his temptation, then in the original Prisoners' Dilemma
game it is the difference y—x but in the Game of Difference it becomes

y— (1 — A,)z — A,oc

ALTRUISM AND SUPERIORITY

117

and it is easily verified that this is always greater than y-x. So each
player's temptation in the Game of Difference is always greater than in
the original game. If we say that a Prisoners' Dilemma becomes 'more
severe' if any player's temptation increases, other things being equal,
then the Game of Difference is a more severe Prisoners' Dilemma than the
original game.

If the untransformed game is a Chicken game, then it is easily shown
that, provided A f #0 (i = 1, 2), the Game of Difference is a Chicken game
or Prisoners' Dilemma according as A; is greater or less than the ratio
(y _ x )/(y _ w ). So the Game of Difference will be a Prisoners' Dilemma -
and thus mutual Cooperation will not occur - if each A, is sufficiently
small, that is, the players attach sufficient weight to achieving superiority
over each other as against their own payoffs per se.

In the special case when A f =0 (i=l, 2), each player seeks only to
maximize the difference between the two payoffs. I call this zero-sum
game a Pure Difference Game. Whatever game we start with, in the Game
of Difference D is the dominant strategy for each player. So (D, D) is the
outcome. But it is no longer Pareto-inferior; for each player is indifferent
between (D, D) and (C, C). 5

(3) Games of Anti-difference or Equality. A player in a Game of Difference
is concerned, to some extent, to raise his own payoff above that of the
other player. An opposite concern could take several forms. The most
extreme of these would presumably be a desire to maximize the excess of
the other player's payoff over one's own. Less extreme would be the
maximization of a convex combination of this excess and one's own
payoff:

".• = A,P; - ( 1 - A,) (p,- - Pj)

with «S A f < 1, but I do not think behaviour of this sort is very
common.

Of more interest, I think, is a desire to minimize the difference, whether
positive or negative, between the other player's payoff and one's own - a
desire, that is, for equality. More generally, consider the game in which
player i seeks to maximize the following utility function:

"i = A I Pi-(l-A i )lpi-p J l
where < A, < 1. (If A f = 1, we have the original Prisoners' Dilemma

118

ALTRUISM AND SUPERIORITY

game.) This represents a convex combination of 'egoism' (maximizing
one's own payoff) and 'equality' (minimizing the absolute value of the
difference between the two payoffs). I call this game a Game of Equality or
Anti-difference. 'Anti-difference' is not a very beautiful expression, but it
is sufficiently neutral to cover a number of different interpretations of
this game.

I will not waste space by setting out the details, but the reader can
easily verify that, unless A, = for either player, both Prisoners' Dilemma
and Chicken games can be transformed into Games of Anti-difference
which are either Prisoners' Dilemma or Chicken games depending on
the values of A;. When A ( = for both players (i.e. in the unlikely event
that both players are concerned with achieving equality regardless of
their own payoffs) the transformed game is a pure coordination game,
each player being indifferent between (C, C) and (D, D).

Sophisticated altruism. An altruistic player was defined earlier as one
whose utility function is of the form :

Ui = Ui(p u p 2 )

where p 1 and p 2 are the payoffs to the two players. A player's utility,
however, may depend not only directly on the other's payoff but also on
his utility. In this case, we have:

Ui = u i (p u p 1 ,Uj), ji=i

Thus, I may derive pleasure directly from the contemplation of your
loss of payoff, but if I believe that you derive pleasure from your loss of
payoff, then I may cease to be happy (and may cease to act so as to
diminish your payoff).

If a player's utility depends in any way on another player's utility, I call
him a sophisticated altruist. 6

Consider, for example, two pure altruists, one positive (A) and the
other negative (B), in a Prisoners' Dilemma. If their utilities are simply of
the form H; = u,(pi, p 2 ), then (C, D) would be the happy outcome (a
Pareto-optimal, unique equilibrium): A would be happy because B's
payoff is a maximum and B would be happy because A's payoff is a
minimum. At the 'first-order of sophistication', A is very happy because
B is happy (with A's minimal payoff), but B is now unhappy because A is
happy (with B's maximal payoff). At the 'second-order of sophistication',

ALTRUISM AND SUPERIORITY

119

A now becomes unhappy because of B's unhappiness at the first-order
level, and B is very unhappy at A's increased happiness at the first-order
level. And so on.

In practice, however, the levels of sophistication will not all carry the
same weight, and there will be an end to this potentially infinite regress.
In particular, it is unlikely that anybody goes beyond the first-order
level:

". = w,(Pi, Pi, Uj(p u p 2 )), j^i

The implications of sophisticated altruism for behaviour in the
Prisoners' Dilemma are illustrated in the following example. Consider a
Prisoners' Dilemma with these payoffs:

1, 1

-1,2

2, -1

(Matrix 1)

Suppose that player 1 is a pure positive altruist while player 2 is a pure
negative altruist, with utilities.

"l=/>2, "2 =

Then the transformed matrix is:

-Pi

1, -1

2, 1

-1, -2

0,0

(Matrix 2)

This is no longer a Prisoners' Dilemma. (C, D) is the Pareto-optimal,
unique equilibrium.

If the players are 'pure, first-order-sophisticated (positive and negat-
ive) altruists', then this matrix in turn becomes:

-1, -1

1, -2

-2, 1

(Matrix 3)

120

ALTRUISM AND SUPERIORITY

which is a Prisoners' Dilemma, but with (C, C) as the only equilibrium
and (D, D) preferred to (C, C) by both players.

If the players are 'pure, second-order-sophisticated (positive and nega-
tive) altruists', then this matrix becomes:

-1,1 -2,-1

1, 2 0,

(Matrix 4)

which is not a Prisoners' Dilemma, (D, C) being the Pareto-optimal,
unique equilibrium. And so on.

More realistically, suppose that both players ignore orders of
sophistication beyond the first, and that their utilities are (for the sake of
illustration )'simply the average of those in Matrix 2 and those at the first
level in Matrix 3; that is,

and

Then the game is:

"i=i/>2+i(-Pi)

«2=£(-Pl)-£(P2)

0,-1 3/2,-1/2
3/2, - 1/2 0,

Here, C dominates D for player 1, while D dominates C for player 2 so
that (C, D) is the only equilibrium. But the game is not a Prisoners'
Dilemma, for this equilibrium is Pareto-optimal.

The supergame. Consider the supergame consisting of an indefinite
number of iterations of the general transformed 2x2 Prisoners'
Dilemma game in which the players combine egoism and unsophisti-
cated altruism. Assume that future payoffs are discounted exponentially,
as in chapters 3 and 4. Then it is a straightforward matter, using the
methods of chapter 3, to derive the conditions for various supergame

ALTRUISM AND SUPERIORITY

121

strategy vectors to be equilibria. It transpires that, in contrast with the
supergame without altruism, in the transformed game every one of the
strategy vectors considered in chapter 3 is an equilibrium for some
values of y, x, w, z, a„ a, and /?, (where a, is i's discount parameter and a,
and ^ are the weights attached by i to his own payoff and to the other
player's payoff, as before). This is not at all surprising: after all, sufficient
positive altruism can transform a one-shot Prisoners' Dilemma into a
game with C dominant for both players; so in the supergame even
(C°°, C°°) is an equilibrium under certain conditions.

In particular, since a Prisoners' Dilemma, when transformed into a
(non-pure) Game of Difference, is still a Prisoners' Dilemma, the results
from chapter 3 can be applied directly. Since the new Prisoners'
Dilemma is 'more severe' than the original one, we should find, in
particular, that the conditions for mutual conditional Cooperation to be
an equilibrium are more stringent. This is easily demonstrated. Recall
that (B, B) is an equilibrium in the untransformed game if and only if

v— x

a, > ~ (5.4)

y—w

(which is the condition for unilateral defection to Z) 00 not to pay) and

y — x

a, > y - (5.5)

x — z

(which is the condition for unilateral defection to B' not to pay). In the
transformed game, the Game of Difference, the equivalent of (5.4) is
easily found to be:

^ y-x + {\-Xj){x-z)
y- w + (l -A,)(w-z)

Since x — z > w-z and 1 - 1, > 0, the right hand side of this exceeds
(y— x)/(y — w); that is, the discount parameter must be greater (the
discount rate must be smaller) in the Game of Difference than in the
original Prisoners' Dilemma if unilateral defection to from (B, B) is
not to pay. Similarly, in the transformed game the equivalent of (5.5) is:

^ y-x + d-k^x-z)

x -z+(i-My-x) (5 - 7)

122

ALTRUISM AND SUPERIORITY

Since x — z > y — x (if it were not, the right hand side of (5.5) would be
greater than one) and 1— A, > 0, the right hand side of (5.7) exceeds
(y — x)/(x - z); that is, the discount parameter must be greater (the
discount rate must be smaller) in the Game of Difference than in the
original Prisoners' Dilemma if unilateral defection to B' from (B, B) is
not to pay.

Thus, the conditions (5.6 and 5.7) for mutual conditional Cooperation
to be an equilibrium in the Game of Difference are indeed more stringent
than they were in the original, untransformed Prisoner' Dilemma.

An AT-person Game of Difference

In the general iV-person Prisoners' Dilemma game, the number of
different forms which altruism can take is very large. I shall consider only
one of them. It is one of many possible generalizations of the two-person
Game of Difference which I discussed earlier. Of all the transformed
Prisoners' Dilemma games involving egoism and some form of altruism,
Games of Difference are the most important for my purposes in this
book. When we come to consider in the next chapter the political theory
of Hobbes, we shall see that the 'game' which Hobbes assumes people to
be playing in the absence of government (in 'the state of nature') is in
effect a generalized Game of Difference.

In the two-person Game of Difference, each individual's utility is a
convex combination of his own payoff (p,) and the difference (pi — pj)
between his and the other player's payoff. Let us anticipate a Hobbesian
term of the next chapter and call this difference the eminence of the i' ,h
over the / h individual. In a game with more than two players, there are a
number of ways in which an individual might be said to seek 'eminence' :
he might, for example, seek to maximize the number of other individuals
with respect to whom he is eminent (his positive eminence); he might
seek to maximize the sum of his eminences over each other individual;
and so on. The definition which I shall adopt here is that an individual's
eminence in the iV-person game is the average of his eminence over each
other individual. I use the same word, eminence, in both the two-person
and N-person cases : eminence in the former is a special case of eminence
in the latter. Thus, the i" lh individual's eminence is denned as :

ALTRUISM AND SUPERIORITY

123

The untransformed game (with payoff p, to the i' ,h individual) is the N-
person Prisoners' Dilemma specified in chapter 4; that is, the two payoff
functions f(v) and g(v) - which are each player's payoffs when he
chooses C and D, respectively, and v others choose D - satisfy the three
conditions:

(i) g(v) > f(v) for all v ^

(ii) /(JV-l)><?(0)

(iii) g(y) > 0(0) for all v >

The TV-person generalization of the two-person Game of Difference
which I shall now consider is the game in which each individual is
assumed to seek to maximize a convex combination of his own payoff
and his eminence. The i' h individual's utility, then, is defined as:

"i = /W>i + (l -Ai)£ ;

where 0^1,^ 1. When 1, = 1 for all i, we have of course the original,
untransformed Prisoners' Dilemma in which every player is a pure
egoist.

I shall do no more than establish the conditons under which this N-
person Game of Difference is a Prisoners' Dilemma game satisfying the
three conditions listed above. This is all I require for my purposes in
chapter 6. For when these conditions are met, the general results on the
N-person Prisoners' Dilemma apply to the Game of Difference.

If the i' th individual and v others Cooperate, then the payoff to i and to
each of the v others is/(v) while the payoff to each of the N - v - 1 who do
not Cooperate is g(v + 1). Thus, individual i's utility is:

= V(v) + ( 1 - v,)jj(N - v - 1 ) {/(v) -g(v+l )}
= F(v), say.

If individual i does not Cooperate, while v others do, then we find that

u, = b9(v) + (1 - v,)^(v){0(v) -/(v - 1 )} if v *
= G(v), say

and

M , = ^(0) = G(0) ifv=0

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ALTRUISM AND SUPERIORITY

because

f(v- l)=/(- 1) is undefined.

The necessary and sufficient condition for the i' h individual to choose
not to Cooperate is of course G(v) > F(v). This condition cannot be
essentially simplified unless further assumptions are made about/(v) and

However, we can certainly say that G(v) > F(v) if (but not only if)

v{<7(v)-/(v-l)} > -(N-V-1){0(V + 1)-/(V)}

which in turn is true if g(v) is strictly increasing with v (for we have
already assumed that g{y) > f(v) for all v).

Thus, if g(v) is strictly increasing, each player prefers D to C no matter
what the other players choose; and therefore the outcome is that every
player chooses D. Let us see whether this outcome is Pareto-inferior.

If everybody Cooperates, each player's payoff is/(JV — 1 ) and therefore
the i' h player's utility is F(N- l) = Xif(N- 1).

If nobody Cooperates, each player's payoff is g(0) and therefore the f th
player's utility is G(0) = ^(0).

As long as Aj#0, we have Aif{N — l)> 1,^(0), for we have already
assumed that f(N — l)> g(0). Thus, if Aj#0 for all f, every individual
prefers the outcome when everybody Cooperates to the outcome when
everybody Defects.

Finally, observe that G(v) > G(0) if (but not only if) g(v) is strictly
increasing with v.

Thus, this N-person Game of Difference is a Prisoners' Dilemma
satisfying conditions (i), (if) and (iff) if (but not only if) A,^0,/or all i, and
g(v) is strictly increasing with v.

The condition A, #0, for all i, simply requires that the game is not one
of pure eminence; each individual must attach some weight to his own
payoff per se. All other convex combinations of his own payoff and his
eminence are permitted.

The condition that g(v) is strictly increasing with v requires only that
the greater the number of individuals who Cooperate, the greater is the
payoff to any individual who does not Cooperate.

6. The state

At the start of this book a sketch was made of an argument for the
desirability of the state. The first part of the argument is that the static
preferences of individuals amongst alternative courses of action with
respect to the provision of public goods (in particular, domestic peace
and security and environmental public goods) are those of a Prisoners'
Dilemma game, at least where relatively large numbers of individuals are
involved. I examined this part of the argument in chapter 2 and found
that it is not necessarily true and needs to be qualified. Accepting the first
part of the argument, I considered in chapters 3 & 4 the second part of
the argument : that individuals would not voluntarily cooperate in such
situations. If the problem is properly specified as a one-shot Prisoners'
Dilemma game, this conclusion is obviously correct: but to treat the
problem, as it is usually treated (tacitly or explicitly), in terms of a one-
shot game is clearly inadequate. I therefore treated it in terms of a
Prisoners' Dilemma supergame in which the players discount future
payoffs. Formulated in this way, the second part of the argument is not
necessarily true: under certain conditions, it is rational to Cooperate in
the supergame. It remains to examine the third part of the argument, that
the failure of individuals, at least in large groups, to cooperate
voluntarily (to provide themselves with certain public goods) makes the
state desirable.

This part of the argument will be examined in the present chapter in
the forms in which it appears in the political theories of Hobbes and
Hume, and more generally in the final chapter.

In this chapter, I consider two versions, those of Hobbes and Hume, of
the whole of the argument about the desirability of the state. My reasons
for doing so, and in particular for choosing Hobbes and Hume as
exemplars of the general argument, were given earlier (in chapter 1 ) and I

125

126

THE STATE

shall not repeat them here. I shall give an account of the two theories in
such a way that they can be compared at certain points with some of the
ideas in the earlier chapters. A large part of my account of Hobbes's
theory is devoted to showing (what at first sight may appear almost
obvious) that men, in what Hobbes calls the 'state of nature', find
themselves in a Prisoners' Dilemma game.

My treatment of Hobbes's theory is based entirely on his Leviathan. I
have resisted the temptation to buttress my argument at any point with
selective quotation from his other works. Leviathan contains the clearest
and most coherent version of the argument which is of interest here, and
I have thought it unjustified to refer to a passage from another version
which, taken as a whole, is different, less coherent and generally less
satisfactory than the one in Leviathan. My treatment of Hume's theory is
based chiefly on A Treatise of Human Nature, which I think gives a
clearer and more complete account than the one in An Enquiry
Concerning the Principles of Morals. The second differs in places from,
but is not inconsistent with, the first, and I have referred to it once or
twice.

Hobbes's Leviathan

I begin with Hobbes's description of those parts of the structure of
individual preferences on which his political theory is based. His
conclusions on this subject are presented here as assumptions, whereas
in Leviathan Hobbes claims to deduce them from more fundamental,
physical premises. His political theory is unaffected by this shift in the
point of logical departure.

Individual preferences

All men, says Hobbes, desire certain things (Lev 31 J. 1 He derives this
proposition from his assumptions about 'motion' and these lead him to
speak of man's ceaseless striving for the things he desires : 'Life it self is
but Motion, and can never be without Desire' (Lev 48); and 'Nor can a
man any more live, whose Desires are at an end, than he whose Senses
and Imagination are at a stand. Felicity is a continuall progresse of the
desire from one object to another; the attaining of the former, being still
but the way to the later' (Lev 75). But this adds nothing to the original

THE STATE

127

statement that all men desire certain things. For this statement, in which
there is no reference to time, is to apply at each point in time. The same is
true of man's 'perpetuall and restlesse desire of Power'.

Hobbes defines power as follows. 'The power of a Man, (to take it
Universally), is his present means, to obtain some future apparent Good.
And is either Originall, or InstrumentalV (Lev 66). Later he concludes that
he puts '. . . for a general inclination of all mankind, a perpetuall and
restelesse desire of Power after power, that ceaseth only in Death' (Lev
75). This can be viewed either as a part of the initial proposition that men
desire certain things (one of them, then, being power) or, better, as
derivative from it: for if a man desires something, he desires also the
means to obtain it in the future ('. . . anything that is a pleasure in the
sense, the same also is pleasure in the imagination' (Lev 76) - although
Hobbes is never explicit about how men presently value future expected
goods).

Thus power-seeking (which has been so much emphasized in discus-
sions of Hobbes) does not play an independent role in Hobbes's theory
and will not appear in my restatement of it.

Now Hobbes seems to say that men do not simply desire certain
primary goods, but rather they desire to have them to an 'eminent'
degree. 'Vertue generally, in all sorts of subjects, is somewhat that is
valued for eminence ; and consisteth in comparison. For if all things were
equally in all men, nothing would be prized' (Lev 52). Thus '. . . man,
whose Joy consisteth in comparing himself with other men, can relish
nothing but what is eminent' (Lev 130). Power, which all men seek and
which is a means to other desirable things, is divided by Hobbes into
'natural' and 'instrumental' power, the first being defined as 'the
eminence of the Faculties of Body, or Mind : as extraordinary Strength,
Forme, Prudence, Arts, Eloquence, Liberality, Nobility', while 'In-
strumentall are those Powers, which acquired by these, or by fortune, are
means and Instruments to acquire more : as Riches, Reputation, Friends,
and the secret working of God, which men call Good Luck' (Lev 66). A
desire for power, then, entails by definition a desire for 'eminence'.

In my restatement of Hobbes's main argument, there will be just one
assumption about individual preferences, to the effect that each person
seeks to maximize a convex combination of his own payoff and his
eminence (to use the language of chapter 5). But the desire for eminence
needs to be specified more carefully than Hobbes does, in the passages

referred to above. No problems arise if there are only two individuals
who both desire the same object and nothing else. For then the
assumption is that each individual seeks to maximize the excess of his
amount of the object over the other individual's amount.

Yet, in the first place, Hobbes has said that not all men desire the same
things (Lev 40). Perhaps, in the two-person case, each man seeks to have
more of the things he wants than the other man has of those same things,
even though the latter has no desire for them. Or, more plausibly, he
seeks to have more of the things he wants than the other man has of the
things he wants : in this case, he must presumably have some means of
comparing his and the other individual's amounts of the different
objects. However, neither of these is terribly plausible unless there is an
acknowledged scheme for comparing the extents to which the two men
desite their different objects. Yet it can be argued that Hobbes has in
mind particular objects of desire and assumes that all men desire at least
these objects. For example, all men desire their own preservation. All
men (he seems to say in Chapter 8) desire to be eminent in the
'intellectual virtues'. All men desire to be eminent in 'Strength, Forme,
Prudence, Arts, Eloquence, Liberality, Nobility', for eminence in these
things gives one 'Natural Power', which all men desire, and all men
desire 'Riches, Reputation, Friends, and the secret working of God,
which men call Good Luck' for these are 'Instrumentall powers', which
all men desire (Lev 66).

It is not necessary here to resolve this problem in Leviathan (if it is a
problem). In whatever way it is resolved, the following assumption (or
something very like it) must in any case be made : corresponding to every
possible state of affairs which is the outcome of individual choices, there
is for each individual a payoff; each individual's payoff scale is unique at
least up to a linear transformation, and each individual is able to
compare other individuals' payoffs with his own (he is able to place any
other player's payoff scale in a one-to-one correspondence with his own).
Then, in the two-person case, if the payoff to the i ,h individual is p u I
define the eminence of the i ,h over the / h individual as p, — pj.

In the second place, there are of course more than two people in the
societies Hobbes is writing about. In this case, there are a number of
ways of specifying the assumption that men seek 'eminence'. (I
mentioned a few in chapter 5.) In Hobbes's theory, several of these
alternative definitions of 'eminence' would suffice as part of the logical

basis for the rest of his argument. For the sake of concreteness, in my
restatement of his argument I shall adopt the definition given in chapter
5 ('An AT-person Game of Difference'), where an individual's eminence is
defined as the average of his eminences over each other individual. Also I
shall assume that each individual seeks to maximize a convex combi-
nation (as defined in the same section) of his own payoff and his
eminence. This convex combination is referred to as his utility. It should
be noted that 'pure egoism' (that is, maximizing one's own payoff only) is
a special case of maximizing this convex combination; in this case,
'utility' and 'payoff can be viewed as the same thing.

In Leviathan (but not in all his earlier works), Hobbes clearly believes
that 'benevolence', 'pity' and other manifestations of positive altruism are
possible, that in some degree they are found in some individuals, and that
they are not reducible to or mediated by self-interest. Nevertheless, it is
true that the assumption on which his political theory is based is that in
the state of nature (that is, in society without government) a man seeks
only to maximize a convex combination of his own payoff and his
eminence; that is to say, his preferences contain a mixture of egoism and
negative altruism only.

A Prisoners' Dilemma

In this section I argue that in what Hobbes calls the 'state of nature' men
find themselves in a Prisoners' Dilemma; that is to say, Hobbes is
assuming that the choices available to each man (or 'player') and the
players' preferences amongst the possible outcomes are such that the
game is a Prisoners' Dilemma; and the Prisoners' Dilemma is the only
structure of utilities (out of a very large number of possibilities) which
Hobbes must have assumed to obtain in the state of nature.

I shall present two versions of this argument and evaluate their
relative merits. The first version argues that Hobbes's theory is
essentially static, being an analysis of the Prisoners' Dilemma ordinary
game. The second version is more dynamic: although, in this version,
time does not play an explicit role and there is no talk of the present
valuation of future benefits, conditional cooperation is thought to be
sometimes rational (and conditional cooperation is of course not
possible in a game played only once). 2

I assume (as Hobbes does in effect) that each individual is confronted
with a number of alternative courses of action, which I call strategies.

130

THE STATE

The number of strategies available to each player is assumed to be just
two. I shall show later that nothing essential in Hobbes's argument is
affected if this assumption is relaxed. Call these two strategies C and D.
(It need not be assumed that C and D denote the same two courses of
action for every individual; but no confusion will arise if the same two
labels are used for all individuals.) There are thus four possible states of
affairs or outcomes. A strategy vector is defined as a list (an ordered N-
tuple, if there are N individuals in the society) of strategies, one for each
player.

One of the outcomes is called 'the state of War' or simply 'War' (Lev
96). This is the state of affairs which obtains when every individual seeks,
in the absence of restraint, to maximize his utility (as Hobbes assumes he
does). If the state of War is to be a determinate, unique outcome, then it
must be assumed that 'maximizing utility' has a clear meaning and
entails the choice by each individual of a single strategy. Let us suppose
that this strategy is D. (For the time being, strategy C is simply 'not
acting without restraint so as to maximize one's utility'.) Now, when men
are under no restraint, when they 'live without a common power to keep
them all in awe', they are said to be in the 'state of nature' {Lev Chapter
8).

When men are not in the state of War, then there is 'Peace', says
Hobbes. There are of course three outcomes other than War (assuming
that there are only two strategies available to each player), but it is clear
that Hobbes means that Peace corresponds to only one of these
outcomes : it obtains only when no individual chooses strategy D. (For he
later argues that everybody must behave differently if society is to move
out of the state of War into that of Peace.) Thus, Peace obtains when
every individual chooses strategy C.

Now the state of War is Pareto-inferior: every man prefers Peace to
War. For in War, 'men live without . . . security' and there is 'continuall
feare, and danger of violent death; And the life of man, solitary, poore,
nasty, brutish, and short' (Lev 96-7). Despite this rhetorical flourish,
Hobbes makes it clear that 'the nature of War, consisteth not in actuall
fighting; but in the known disposition thereto, during all the time there is
no assurance to the contrary' (Lev 96). Nevertheless, in this condition, a
man cannot expect to obtain what he desires; whereas in the state of
Peace, life and security are guaranteed to each man and he can
reasonably expect to obtain some of the things he desires; so that 'all men
agree . . . that Peace is good' (Lev 122).

THE STATE

131

In the state of nature, then, each man will so act that the outcome is
War, which is Pareto-inferior. The only way, in Hobbes's view, to
prevent this outcome occurring and to ensure Peace instead is for men to
erect a 'common power' which will maintain conditions in which each
individual will not wish to choose D; and the only way to erect such a
common power is to 'authorize', 'by covenanting' amongst themselves,
one man or assembly of men to do whatever is necessary to maintain
such conditions. The man or assembly of men so authorized is called the
'Sovereign' (Lev Chapter 17).

This needs explanation. For this, several definitions are required. The
'right of nature', says Hobbes, is 'the Liberty each man hath, to use his
own power, as he will himselfe, for the preservation of his own Nature;
that is to say, of his own Life; and consequently, of doing any thing,
which in his own Judgement, and Reason, hee shall conceive to be the
aptest means thereunto' (Lev 99). In the state of nature, therefore, 'every
man has a Right to every thing ; even to one anothers body' (Lev 99); so
there is no security of life; 'and consequently, it is a precept, or generall
rule of Reason, That every man, ought to endeavour Peace, as farre forth as
he has hope of obtaining it ; and when he cannot obtain it, that he may seek,
and use, all helps, and advantages of Warre' (Lev 100). Notice that the
second part of this statement is just the Right of Nature and that this is
part of 'a precept, or generall rule of Reason'. Hobbes is saying, in effect,
that it is rational for a man to choose D if he thinks he cannot obtain
Peace, that is, if he thinks that some other people will choose D. Hobbes
calls the first part of this statement the Fundamental or First Law of
Nature, a 'law of nature' having been defined earlier as 'a Precept, or
generall Rule, found out by Reason, by which a man is forbidden to do,
that, which is destructive of his life, or taketh away the means of
preserving the same; and to omit, that, by which he thinketh it may be
best preserved' (Lev 99).

We may now say that strategy C is 'laying aside one's Right of Nature'.
This only restates our earlier definition of C as not acting without
restraint so as to maximize one's utility.

This statement of the first Law of Nature will be clarified after we have
seen Hobbes's discussion of obligation and covenanting.

The definition of obligation given in Leviathan is straightforward. The
second Law of Nature requires a man under certain conditions to lay
down his right to all things. He may do this by simply 'renouncing' it
('when he care not to whom the benefit thereof redoundeth') or by

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'transferring' it to another (when he does so care). 'And when a man hath
in either manner abandoned, or granted away his Right, then he is said to
be obliged, or bound, not to hinder those, to whom such Right is
granted, or abandoned, from the benefit of it' (Lev 101). A man will of
course only transfer or renounce his right in exchange for some good to
himself, in particular 'for some Right reciprocally transferred to himselfe'
(Lev 101). This mutual transferring of right is called 'contract', and a
contract in which at least one of the parties promises to perform his part
in the future is called a 'covenant' (Lev 102), a 'covenant of mutual trust'
being one in which both parties so promise (Lev 105, 1 10).

Now contracts would not serve their end of securing Peace if they were
not kept. Thus, the third Law of Nature is that 'men performe their
Covenants made' (Lev 1 10). Yet, by performing unilaterally his part of a
covenant of mutual trust, a man may expose himself, and this he is
forbidden to do by all the Laws of Nature. Thus, a man should not do his
part unless he is sure that the other party will do his. A covenant of
mutual trust made in the state of nature is therefore void 'upon any
reasonable suspicion': 'For he that performeth first, has no assurance
that the other will performe after . . . And therefore . . . does but betray
himselfe to his enemy; contrary to the Right (he can never abandon) of
defending his life, and means of living'. But 'if there be a common Power
set over them both, with right and force sufficient to compell perform-
ance', then 'that feare is no more reasonable' and so the covenant is not
void (Lev 105).

Thus Hobbes comes to the last stage of his main argument. The central
point (leaving aside the details of 'authorization' and of acquisition of
sovereignty by conquest) is that the only way for men to obtain Peace is
for every man to make with every other man a covenant of mutual trust
instituting a Sovereign with the power to do whatever is necessary to
secure Peace (Lev 131-2). The Sovereign will maintain Peace by
compelling men 'equally to the performance of their Covenants, by the
terrour of some punishment, greater than the benefit they expect by the
breach of their Covenant' (Lev 110); in other words, by creating
appropriate laws and punishing transgressors.

I will return later to this point in Hobbes's argument to discuss his
account of why men obey the Sovereign, and again on pp. 146-8 to
present his description of the Sovereign's powers.

We are now in a position to develop the argument that in the state of
nature men find themselves in a Prisoners' Dilemma game.

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The 'game' which Hobbesian men in the state of nature are 'playing' is
certainly non-cooperative (in the game theorists' sense, as I defined it in
chapter 1), for although agreements are possible, they are not binding:
the state of nature, by definition, is precisely the absence of any
constraint which would keep men to their agreements.

It has been assumed that each man has a choice between two
strategies, C and D. We have already seen that in the state of nature it is
rational for each man to choose D, and thus the outcome in the state of
nature is War. Since, as we have seen, the state of Peace is preferred by
every individual to the state of War, we can conclude that the game is a
Prisoners' Dilemma if D is the 'rational' strategy for each man, in the
sense that it dominates all other strategies, that is, it yields a more
preferred outcome than any other strategy no matter what strategies the
other individuals choose. (There are of course other and more con-
troversial ways in which a strategy can be said to be the 'rational' one to
use; but dominance is required for the game to be a Prisoners' Dilemma.)

Now first, it is clear that in the state of nature no individual has an
incentive unilaterally to change his strategy from D to C, if the other
players are choosing D. For, as we have seen already, 'if other men will
not lay down their Right, as well as he; then there is no Reason for any
one, to devest himselfe of his: For that were to expose himselfe to
Prey . . .' (Lev 100). If this were not the case, there would be no need for
him to enter into a covenant. Thus the state of War, (D, £>,...,£>), is
certainly an equilibrium.

Consider next any two individuals who have made a covenant of
mutual trust to lay down their Right of Nature, that is, to choose C.
Hobbes says that it does not pay a man to perform his part of the
covenant (to choose £)) if he believes the other man will not. On each of
the occasions in Leviathan where he argues this, there is no mention of
what all the other members of the society are doing. Yet, clearly, the
payoffs to the two individuals, each of whom is choosing between
keeping and not keeping the agreement, depend on what the rest of the
society is doing (on how many others are choosing D, for example). We
must infer that Hobbes is assuming that his argument holds no matter
what others are doing. It follows that, just as long as one other individual
(the one with whom I am covenanting) chooses D, it pays me to choose D
also.

This establishes that D dominates C for each individual in every
contingency (i.e. for every combination of strategy choices by the JV — 1

other individuals) except where all other individuals choose C. This
contingency remains to be considered. And it is here that I believe
Hobbes's argument is not wholly satisfactory; it is this contingency
which gives rise to the two interpretations I mentioned earlier. The first
interpretation is the static one, that Hobbes treats a Prisoners' Dilemma
game played only once. The second interpretation is more dynamic, at
least to the extent that it admits sequences of choices and the possibility
of using conditional strategies (which are of course ruled out in a one-
shot game).

Now if, as I believe, individual preferences in Hobbes's state of nature
have the structure of a Prisoners' Dilemma game at any point in time,
then (i) if the first interpretation is the correct one, it will never be
rational (in the state of nature) for any individual to choose C, even if
(indeed, especially if) every other individual chooses C; it will never be
rational for a party to a covenant to keep his promise if the other party
has performed his part already (and it makes no difference, in the state of
nature, whether the two players make their choices simultaneously or
one player's choice follows the other's in full knowledge of it); but (ii) if
the second, dynamic interpretation is the correct one, it may be rational
for an individual to Cooperate when the other individuals Cooperate;
more precisely, we have seen (in chapters 3 and 4) that conditional
Cooperation is rational under certain conditions in a (two-person or N-
person) Prisoners' Dilemma supergame with future payoffs exponen-
tially discounted. With this in mind, let us examine the relative merits of
the two interpretations.

Hobbes does in fact assert that if one of the parties to a covenant has
already performed his part, then it is rational (and obligatory) for the
other to perform his, even in the state of nature: '. . . where one of the
parties has performed already; or where there is a Power to make him
performe; there is the question whether it be against reason, that is,
against the benefit of the other to performe, or not. And I say it is not
against reason' (Lev 112). This statement would seem to preclude the
first, static interpretation, or at least to render it less plausible. In the
continuation of this passage, Hobbes explaines why it is 'not against
Reason' to Cooperate when others do :

First, that when a man doth a thing, which notwithstanding any thing
can be foreseen, and reckoned on, tendeth to his own destruction,
howsoever some accident which he could not expect, arriving may

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turne it to his benefit; yet such events do not make it reasonably or
wisely done. Secondly, that in a condition of Warre, wherein every
man to every man, for want of a common Power to keep them all in
awe, is an Enemy, there is no man can hope by his own strength, or
wit, to defend himselfe from destruction, without the help of
Confederates; . . . and therefore he which declares he thinks it reason
to deceive those that help him, can in reason expect no other means of
safety, than what can be had from his own single Power. He therefore
that breaketh his Covenant, . . . cannot be received into any Society,
that unite themselves for Peace and Defence, but by the errour of
them that receive him; nor when he is received, be retayned in it,
without seeing the danger of their errour; which errours a man cannot
reasonably reckon on as the means of his security. (Lev 112)

In this passage, Hobbes clearly suggests that a man should perform his
part of a covenant after the other has done so, out of a fear of the future
consequences to himself should he not do so; in other words, it is
suggested that the behaviour of each of the two parties to the covenant is
conditional upon the behaviour of the other. The same idea appears in
an earlier passage (Lev 108-9) where Hobbes mentions two other
possible motives for not breaking any sort of covenant (not merely one in
which the other party has already performed). These are fear of the
consequences of breaking one's word and pride in appearing not to need
to break it. The latter is a 'generosity too rarely found to be presumed
upon'. The former is of two kinds: fear of the power of those one might
offend and fear of God. The first of these is too limited to be effective,
because of the approximate equality (to be discussed shortly) of men in
the state of nature. This leaves only the fear of God ; but Hobbes sets little
store by this and clearly thinks it will not be effective enough to keep men
to their covenants.

This idea of conditional cooperation is expressed more generally in
the Fundamental Law of Nature, That every man, ought to endeavour
Peace, asfarre as he has hope of obtaining it; and when he cannot obtain it,
that he may seek, and use, all helps, and advantages of Warre', and in the
Second Law of Nature which follows from it, 'That a man be willing,
when others are so too, as farre-forth, as for Peace, and defence of himself
he shall think it necessary, to lay down this right to all things; and be
contented with so much liberty against other men, as he would allow
other men against himselfe' (Lev 100; emphasis supplied - the originals

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are italicized throughout). Hobbes seems to be saying here, and in the
passages quoted earlier, that every man ought always to do what is
conducive to Peace just as long as he can do so safely and this means
that, in the state of nature, he should Cooperate if others do, but
otherwise he should not Cooperate. This sounds like the 'tit-for-tat'
strategy B or its Af-person generalization B n which were considered in
chapters 3 and 4, though it obviously cannot be said that this is precisely
what Hobbes had in mind.

Earlier, we saw that the Sovereign, which men institute by covenants
of mutual trust, will maintain Peace by compelling men 'equally to the
performance of their Covenants, by the terrour of some punishment,
greater than the benefit they expect by the breach of their Covenant' (Lev
1 10). Yet it is clear that Hobbes is not simply asserting that the Sovereign
will be effective in maintaining Peace because each man will obey him
only from fear of his sanctions. He believes that because men want Peace
(or at least prefer Peace to War), each of them will obey because the
Sovereign has removed the only reason for not keeping one's covenants,
which is a 'reasonable suspicion' that other men will not keep theirs.

Hobbes's position here is widely misunderstood. A standard view of
Hobbes is that 'he has such a limited view of human motives that he
cannot provide any other explanation for acceptance of authority than
the fear of . . . sanctions'. 3 Exceptions to this distortion of what Hobbes
actually said in Leviathan are rare. H. L. A. Hart, in his discussion of the
'minimal content of natural law', based on Leviathan and on Hume's
Treatise, concludes that centrally organized sanctions are required 'not
as the normal motive for obedience, but as a guarantee that those who
would voluntarily obey should not be sacrificed to those who would
not'; 4 and Brian Barry writes that 'It is not so much that the Sovereign
makes it pay to keep your covenant by punishing you if you don't, but
that it always pays anyway to keep covenants provided you can do so
without exposing yourself. 5 The Sovereign is required, then, to ensure
that nobody will expose himself.

Let us recall the results of the analysis of JV-person Prisoners'
Dilemma supergames in Chapter 4. It was found there that Cooperation
in these games is rational only under certain conditions : in the first place,
only conditional Cooperation is ever rational, and it must be contingent
upon the Cooperation (in the previous constituent game) of all the other
Cooperators (conditional and unconditional); in the second place, the

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discount rates of each of the Cooperators must not be too high, relative
to a certain function of the constituent game payoffs. (The first condition
does not require Cooperation to be conditional upon the Cooperation of
all the N - 1 other players; it can be rational for each of a subset of the N
players to Cooperate conditionally while the remaining players use
unconditionally Cooperative or non-Cooperative strategies.) Now, we
obviously cannot make a precise comparison of Hobbes's argument with
these results. Hobbes does not clearly specify the form of the conditional
Cooperation which he says is rational in the state of nature (is it, for
example, contingent upon other players' behaviour in only the im-
mediately preceding time period?); there is no talk of discounting of
future benefits ; and so on. Nevertheless, it is clear from these results that,
even if the requirement concerning the discount rates is ignored,
voluntary Cooperation in the N-person Prisoners' Dilemma supergame
is somewhat precarious, and it can be argued that it was just this
precariousness which in Hobbes's view made a Sovereign necessary: the
Sovereign would provide the conditions in which it was rational for a
man to Cooperate conditionally, by ensuring that he could rely on a
sufficient number of other individuals to Cooperate.

This, then, is the case for the second, dynamic interpretation of what I
take to be the core of Hobbes's political theory. According to this
interpretation, individual preferences at any point in time are those of a
Prisoners' Dilemma; nevertheless it is rational to Cooperate condition-
ally. The problem with this interpretation, however, is that, although the
idea of conditional Cooperation is in Leviathan (and therefore Hobbes's
analysis cannot be entirely static, for conditional Cooperation is not
possible in a game played only once), Hobbes has virtually nothing
explicit to say about any sort of dynamics. Time plays no explicit role in
his political theory. It is true that Hobbes sometimes speaks of
'anticipation' and 'foresight' and of how men are in 'a perpetuall
solicitude of the time to come', but on these occasions he is not speaking
of the present valuation of future benefits and the effect of discounting on
the prospects for voluntary Cooperation. Nor do his few explicit
statements on the subject of discounting play an essential role in this
theory. The most explicit statement of this kind is where, speaking of the
unwillingness of the Sovereign's subjects to pay their taxes so that he
may be enabled to defend them at any time in the future, Hobbes says:
'For all men are by nature provided of notable magnifying glasses, (that

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is their Passions and Selfe-love), through which, every little payment
appeareth a great grievance; but are destitute of those prospective
glasses (namely Morall and Civill Science,) to see a farre off the miseries
that hang over them, and cannot without such payment be avoyded'
(Lev 141). This preference of man for a near to a remote good plays an
important role in Hume's justification of government, as we shall see
later, but nothing is made of the idea in Hobbes's Leviathan.

It is for this reason that it is tempting to fall back on the first, static
interpretation: that Hobbes is in effect treating only a Prisoners'
Dilemma ordinary game, with no dynamic elements at all. Yet on this
view Hobbes's theory is not entirely coherent. Most of what he says is
certainly consistent with the view that individual preferences are those of
a Prisoners' Dilemma; but, as we have seen, Hobbes argues that it is
rational for an individual to choose C if the other players do, whereas, of
course, it is not rational in a Prisoners' Dilemma game to choose C in
any contingency, if that game is played only once. It seems to me, then,
that the more dynamic interpretation, in which conditional Cooperation
is rational (always rational according to Hobbes, though only sometimes
rational in the supergame model of chapters 3 and 4) but precarious, is
closer to what Hobbes has to say in Leviathan, but at the same time it has
to be admitted that Hobbes does not give a very full account of any sort
of dynamics of interdependent individual choices.

There is a shorter route which might be taken to the conclusion that
Hobbes is talking about a Prisoners' Dilemma one-shot game than the
one taken at the start of this section, and it does not involve any
consideration of the performance of covenants. Hobbes says : 'Feare of
oppression, disposeth a man to anticipate, or to seek ayd by society: for
there is no other way by which a man can secure his life and liberty' (Lev
77); and again, because of the 'diffidence' which every man has, simply by
virtue of his knowledge that others, like himself, are seeking to maximize
their utility, 'there is no way for any man to secure himselfe, so
reasonable, as Anticipation . . .' (Lev 95). Now Hobbes could be read as
asserting here that a man should choose D because he can be fairly sure
that the others will choose D, and even if they do not, D is still his best
strategy. (And it makes no difference whether the others choose at the
same time as he does, or at a later time with or without knowledge of his
choice.) However, this is perhaps reading too much into too little. In any
case it would still have to be shown that the remainder of the core of

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Hobbes's argument was consistent with the assumption that the game is
a Prisoners' Dilemma played only once.

It is worth noting here that if it were accepted that in the state of nature
men find themselves in a Prisoners' Dilemma one-shot game, then it
would not make sense to argue that the Sovereign's sanctions are
required, not so much to compel everybody to obey, but rather to
provide a guarantee that those who would obey voluntarily can do so
without exposing themselves. Clearly, if the 'game' in the state of nature
is a Prisoners' Dilemma, then it follows that if a player is certain that the
other players will choose C (because they fear the Sovereign's punish-
ments), then he would not consider it in his interest to choose C himself -
unless he fears the Sovereign's punishments. In other words, although his
expectation that the Sovereign would punish others for their dis-
obedience may reassure him that he will not be 'double-crossed', this
alone does not give him reason to obey. Rather, it gives him a greater
incentive to disobey: unless the Sovereign's presence changes his
(subjective, perceived) utilities as well as his perception of the other
players' utilities.

If it is still insisted that Hobbes is analysing a game played only once,
but this game is not a Prisoners' Dilemma, then there are, I think, only
two plausible alternatives. In both of them, as in the Prisoners' Dilemma
game, each player prefers Peace to War and it pays each player to choose
D if the other players do, for there is no question about these two items in
Leviathan. But in the first alternative game, each player prefers to
Cooperate rather than Defect as long as all other players Cooperate. In
the two-person case, then, the preferences take the following form :

X, X

2, y

y. 2

w, w

with x > y > w > z; whereas in the two-person Prisoners' Dilemma the
utilities satisfied the inequalities y > x > w > z. In this new game, there
is not a dominating strategy for either player and there are now two
equilibria, (C, C) and (D, D). Yet since (C, C) is preferred by both players
to (D, D), neither player will expect (D, D) to be the outcome, so it will not
be the outcome (cf. the discussion of equilibria and outcomes in chapter

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3). In this game there is no need for coercion to prevent a Pareto-inferior
equilibrium occurring; mutual Cooperation will occur without it.

The second alternative game is the one which Hart might have in mind
if he is not thinking of a more dynamic model. Some players' preferences
amongst the possible outcomes are as in the Prisoners' Dilemma game;
those of the others are as in the first alternative game which I have just
defined. In its simplest version, the two-person game in which there is
one player with each of these types of preferences, the preferences take
the following form:

x, x'

y, z'

w, w'

with y > x > w > z and x' > y' > w' > z'. Player 1 (the row-chooser) is
the sort of person who would Cooperate only if coerced, that is, only
through fear of punishment. Player 2 is the sort of person who would
Cooperate as long as the other does too ; he would not take advantage of
the other player. (D, D) is the only equilibrium in this game, and it will
therefore be the outcome. Strategy D is of course dominant for player 1 ;
player 2, seeing this, would also choose D. Thus, it appears that coercion
is necessary to achieve (C, C); the Sovereign will protect player 2 against
player 1 ; he will directly coerce player 1 by threatening sanctions, and he
will thereby provide player 2, who would voluntarily Cooperate if he
could only be sure that others would too, with a guarantee that he will
not expose himself by choosing C.

I think that the assumptions made in this second alternative to the
Prisoners' Dilemma game have some plausibility, but they do not fit very
well with most of what Hobbes says in Leviathan, since they require that
some players have a different sort of preference than the others; whereas
there is very little in Leviathan which does not ascribe the same 'nature'
to all men. Hobbes says, it is true, that some men take 'pleasure in
contemplating their own power in the acts of conquests, which they
pursue farther than their security requires', while others 'would be glad
to be at ease within modest bounds' (Lev 95). But even if this and similar
remarks could be interpreted as meaning that some men would choose C
provided only that others would do likewise, there remains the fact that

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in those statements in Leviathan which I have used to support my
contention that individual preferences at any point in time in the state of
nature are those of a Prisoners' Dilemma game, Hobbes is not speaking
of some people. And if all men would Cooperate as long as others do, as
in the first alternative to the Prisoners' Dilemma, then Hobbes's problem
disappears.

In my discussion of Hobbes's political theory I have not so far
mentioned his assumption of 'equality'. This assumption is that :

nature hath made men so equall, in the faculties of body and mind;
as that though there bee found one man sometimes manifestly
stronger in body, or of quicker mind than another; yet when all is
reckoned together, the difference between man, and man, is not so
considerable, as that one man can thereupon claim to himselfe any
benefit, to which another may not pretend, as well as he. For as to the
strength of body, the weakest has strength enough to kill the
strongest, either by secret machination, or by confederacy with
others, that are in the same danger with himselfe' (Lev 94).

If the assumption is made that the outcomes of the game and the
individuals' preferences amongst them are such that the game at any
point in time is a Prisoners' Dilemma, then the assumption of equality is
superfluous. For it is, in effect, an assertion of strategic interdependence :
that no man alone controls the outcome of the game. No man is safe in
the state of nature; he must fear every other man. The outcome of the
game and therefore his own payoff depend on the actions of all other men
as well as his own. This is the case in a Prisoners' Dilemma game.

Hobbes is not, of course, asserting that the payoffs for each outcome
are the same for all players; this can never be the case in a Prisoners'
Dilemma. Nor is he asserting (as I assumed in chapters 3-5 to simplify
my analysis) that the payoff matrix is necessarily symmetric. It is of
course possible in a Prisoners' Dilemma that the players have very
unequal payoffs for those outcomes in which they all choose C or all
choose D. Hobbes himself clearly did not expect all men to be equally
successful in obtaining what they wanted either in the state of nature,
which is a state of War, or when at Peace under a Sovereign. 6

It remains for me to show that nothing essential in Hobbes's argument
is altered if the number of strategies available to each player is greater
than two. Hobbes himself seems to assume only two strategies; he speaks

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only of 'laying aside one's natural right to all things' or not doing so. But
of course there are degrees to which one may lay aside this right, or
degrees of cooperation. Thus, to use an example of the kind discussed in
chapter 1, if unrestricted pollution of a lake is strategy D, there are
presumably numerous alternatives to D, corresponding to the possible
levels of individual pollution less than D. As before, in the state of nature,
every player chooses D; the resulting outcome is (D, D, . . ., D) which is
Pareto-inferior. The Hobbesian problem remains the same: to get the
players from this 'miserable condition' to an outcome preferred by every
player. If there is only one such outcome this is presumably the only
outcome which the players would covenant to have enforced. Hobbes's
analysis of covenanting applies unchanged to this convenant. Usually,
however, there will be a set (S, say) of outcomes preferred by every player
to (D, D, . . ., D). The players would presumably only consider
covenanting to enforce one of those which are Pareto-optimal with
respect to the set S. A covenant to enforce any one of these would be
necessary and Hobbes's argument applies to each of the possible
covenants. The only new element introduced here is the problem of
agreeing on one of the Pareto-optimal outcomes: of agreeing, for
example, on a particular level of permissible individual pollution.
Hobbes does not of course consider this; but his own analysis, as far as it
goes, applies with full force to this multi-strategy case: men will not
voluntarily act so as to obtain any one of the Pareto-superior outcomes;
they will not keep covenants to refrain from choosing D and use some
other strategy; they must erect a 'common power' with sufficient power
to enforce one of the Pareto-superior outcomes.

A Game of Difference

I argued on pp. 126-9 that the utility which a Hobbesian man seeks to
maximize is a convex combination of his own payoff and his eminence.
Eminence was defined there as the average of his eminence with respect to
each other individual, and his eminence with respect to another
individual was defined as the excess of his payoff over that of the other
individual's.

On pp. 129-42 1 argued that in Hobbes's state of nature the individual
preferences are such that at any point in time the players are in a
Prisoners' Dilemma. The argument was entirely in terms of ordinal
preferences ; that is to say, it was independent of any considerations of

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the relative degree to which one outcome is preferred to another. In
particular, it did not rest on the assumption that the utility of an outcome
to a player takes the form assumed on pp. 126-9 (and is thus a cardinal
utility).

If the game defined in terms of the basic payoffs (the 'basic game') is a
Prisoners' Dilemma, then we know from chapter 5 that the game defined
in terms of the derived utilities (the 'transformed game') is also a
Prisoners' Dilemma, if two conditions are met: (i) i, is non-zero; that is,
the game is not one of pure Difference; and (ii) the payoff g(v) to a player
who chooses D is strictly increasing with the number of other individuals
(v) who choose C.

However, if the transformed game is a Prisoners' Dilemma, it does not
follow that the basic game is a Prisoners' Dilemma. A simple two-person
example shows this: if the payoff matrix is

"2,2 -2,1"
I, -2 1, 1

which is not a Prisoners' Dilemma, then the utility matrix for the Game
of Difference (with X t = ^ for i = 1, 2) is

"1,1 -2i,2"
_2,-2* ii_

which is a Prisoners' Dilemma.

This reveals the possibility that the Hobbesian problem is the result of
man's desire for eminence. (The above example illustrates this : there are
two equilibria in the basic game, but neither player would expect ( 1 , 1 ) to
be the outcome, since both players prefer (2, 2) to it. Thus the outcome is
(2, 2) which is Pareto-optimal, and there is no Hobbesian problem.) It
would be of some interest to discover which sorts of games, not
themselves Prisoners' Dilemmas, become Prisoners' Dilemmas when
transformed to Games of Difference.

I shall not pursue this question, for I believe that in the problems of
interest here (those of the kind discussed in chapter 1 ) the basic ordinary
game is itself a Prisoners' Dilemma. If this is the case, then the
transformed game is also a Prisoners' Dilemma, and this is true no
matter how much 'eminence' relative to 'egoism' we assume (or read into
Hobbes), just as long as eminence is not a man's only concern (that is, as
long as A,- is non-zero) and g(v) is increasing with v.

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The theory restated

I can now recapitulate most of the discussion so far by restating briefly
Hobbes's central argument in Leviathan.

Only three assumptions are necessary. First, that in the state of nature,
men find themselves in a Prisoners' Dilemma; that is, the choices
confronting them and their preferences amongst the possible outcomes
are such that the game which they are playing is at each point in time a
Prisoners' Dilemma. The Prisoners' Dilemma is denned in the usual
way; in particular, it is a non-cooperative game, so that there is nothing
to keep men to any agreements they might make.

There are two versions of the second assumption, corresponding to
the two interpretations of Hobbes's argument put forward on pp. 129-42
above in connection with what Hobbes has to say about the rationality
of Cooperating when others do. The first version is that the Prisoners'
Dilemma game mentioned in the first assumption is not iterated; the
whole theory is restricted to the Prisoners' Dilemma one-shot game. The
second version is that the Prisoners' Dilemma is iterated ; we need not
(and on the basis of what Hobbes actually says, we cannot) go further
than this and say, for example, that the Prisoners' Dilemma game
mentioned in the first assumption is a constituent game of a supergame
with future benefits discounted.

The third assumption is that each individual seeks to obtain an
outcome which is as high as possible in his preference ranking of
outcomes. Equivalently, we may say that each individual seeks to
maximize his utility. In particular, if (as in the Prisoners' Dilemma) he
has a single dominant strategy, he uses it.

In this third assumption, it does not matter what is the basis of a man's
preferences or how 'utility' is defined (as long as the resulting game is a
Prisoners' Dilemma). However, I have argued that Hobbes assumes that
a man's utility is some convex combination of his own payoff and his
eminence.

If the first (static) version of the second assumption is accepted, then it
follows from the three assumptions that the outcome of the game is
(D,D,. . .,£>); that is, the condition of men in the state of nature is War.
This outcome is Pareto-inferior. There is one (and only one) outcome
which every player prefers to it, namely (C, C, . . ., C), which is the state of
Peace. But if the players agreed that each of them should choose C, there

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would be no incentive in the state of nature for any of them to carry out
his part of the agreement. Clearly, if Peace is to be achieved, every man
must be coerced, by which I mean simply that he must be made somehow
to behave differently than he otherwise would (that is to say, than he
would 'voluntarily' in the state of nature).

If the second (dynamic) version of the second assumption is accepted,
then what follows from the three assumptions depends on the precise
form of the dynamic model specified. Hobbes is not sufficiently specific
here, but it is reasonable to conclude (on the basis of the analysis in
chapters 3 and 4) that conditional Cooperation is sometimes rational
(even though not all the other players Cooperate) but rather precarious,
since the Cooperation of each of the conditional Cooperators must be
contingent upon the Cooperation of all the other Cooperators and the
discount rates of every one of the Cooperators must not be too high. It
can be argued that it is this precariousness which in Hobbes's view
makes coercion necessary if Peace is to be achieved, though the necessity
of coercion is clearly less apparent here than in the case when the
Prisoners' Dilemma is assumed not to be iterated.

But Hobbes goes further than this of course. For he specifies in some
detail the particular form that the coercion must take and how it is to be
created. Each man must make a covenant with every other man in which
he promises, on the condition that the other party to the covenant does
likewise, to relinquish the right to all things which he has in the state of
nature in order that a 'Sovereign' may enjoy without restraint his natural
right to all things and thereby be enabled to ensure 'Peace at home, and
mutual ayd against enemies abroad' {Lev Chapter 17). The Sovereign
must be either one man or an assembly of men, though the former is
preferable (Lev 143-7).

Hobbes gives two accounts of how a particular man or assembly of
men is to be made Sovereign. In the first, the Sovereign is specified in the
covenants of each man with every other man and is thus unanimously
agreed on (Lev 132). In the second, there is in effect a unanimous
agreement, in the form of the covenants between every pair of men, to
abide by a majority choice of a particular Sovereign (Lev 133). My
argument is unaffected by this discrepancy; either version may be
chosen.

This, in bare outline (for I have omitted, in particular, any reference to
'authorization') is the 'Generation of that great leviathan' (Lev 132).

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But this Leviathan (whose powers will be described in the next part) is
not the only possible form which the necessary coercion can assume.
One alternative, which is in fact sufficient to maintain Peace (in Hobbes's
sense) in many so-called primitive societies, is the system of controls
characteristic of the small community. (See the brief discussion in
chapter 1.) Hobbes did not discount such possibilities; he believed that
by themselves they would be inadequate. However, if the core of
Hobbes's theory is based, as I have argued, on the assumptions that men
in the state of nature are players in a Prisoners' Dilemma game and that
men are utility maximizers, then, whether the game is iterated or not,
Hobbes cannot legitimately deduce the necessary of any particular form
of coercion, but can only deduce the necessity of any form of coercion
which has the ability, and is seen to have the ability, to deter men from
breaking their covenants.

The Sovereign's powers

Whether Sovereignty has been instituted, in the manner I have just
described, or has been acquired by force, the most important of the
Sovereign's rights and powers are as follows. His subjects cannot change
the form of government or transfer their allegiance to another man or
assembly, without the Sovereign's permission; disagreement with the
majority's choice of a particular Sovereign does not exempt a man from
his obligation to obey the Sovereign; the Sovereign's subjects can neither
'justly' complain of his actions nor 'justly' punish him (this is a trivial
consequence of Hobbes's definitions of justice and authority); the
Sovereign has the right to do whatever he thinks is necessary to maintain
Peace at home and defence against foreign enemies ; he has the right to
judge which opinions and doctrines are to be permitted in public
speeches and publications, as being not detrimental to Peace; he has the
'whole Power of prescribing the Rules, whereby a man may know, what
Goods he may enjoy, and what actions he may do, without being
molested by any of his fellow subjects'; he has the 'Right of Judicature',
that is to say, 'of deciding all controversies' ; he has the right of making
war and peace with other nations and commonwealths, when he thinks it
is 'for the public good', of maintaining an army and taxing his subjects to
pay for it, and (of course) of being in command of it; he has the right to
choose 'all Counsellours, and Ministers, both of Peace and War'; he has

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the right to reward and punish his subjects according to the laws he has
already made, or, in the absence of a law, as he thinks will most conduce
'to the encouraging of men to serve the Commonwealth, or deterring
them from doing dis-service to the same' {Lev chapter 18); and finally,
the Sovereign has the right to choose his successor (Lev 149). These
rights, says Hobbes, are indivisible, for control of the judicature is of no
use without control of a militia to execute the laws, and control of the
militia is of no avail without the right to legislate taxes to support it, and
so on (Lev 139).

This makes the Sovereign very powerful. Hobbes himself sometimes
describes the Sovereign's power as being 'absolute' and 'unlimited' and
'as great, as possibly men can be imagined to make it' (Lev 160).
Nevertheless, it has to be emphasized that Hobbes consistently makes it
clear that the Great Leviathan exists only to maintain Peace amongst his
subjects and to defend them against foreign enemies and that his powers
are only those which are required to perform this role. Thus, in their
covenants with each other to institute a Sovereign, men authorize the
Sovereign to 'Act, or cause to be Acted, in those things which concern the
Common Peace and Safetie' (Lev 131), and by this authority 'he hath the
use of so much Power and Strength conferred on him, that by terror
thereof, he is inabled to forme the wills of them all, to Peace at home, and
mutuall ayd against their enemies abroad' (Lev 132). Again, 'the office
of the Soveraign . . . consisteth in the end, for which he was trusted with
the Soveraign Power, namely the procuration of the safety of the people'
(Lev 258). To this end, he must make 'good laws', a good law being one
'which is Needful, for the Good of the People . . .'; and Hobbes adds that
'Unnecessary Lawes are not good Lawes ; but trapps for Mony . . .' (Lev
268). In the few places where he speaks of the Sovereign's 'absolute
power', he seems to be equating it only with that power which is
'necessarily required' for 'the Peace, and defence of the Commonwealth'
(Lev 247). Above all, he asserts that obedience to the Sovereign is
obligatory only as long as he is doing what he was established for,
namely, maintaining Peace and defence (Lev 170).

I have argued in the preceding section that Hobbes may not
legitimately deduce from his own assumptions the conclusion that the
coercion which is necessary to get men out of the condition of War must
take the particular form which he specifies. However, if the coercion
must be in the form of a Sovereign which is either one man or an

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assembly of men, then Hobbes is quite correct to give the Sovereign just
those powers which are required by him to maintain Peace. I have
argued in this part that this is what Hobbes does.

Possessive market society

C. B. Macpherson, in his widely read book, The Political Theory of
Possessive Individualism, has put forward a reconstruction of Hobbes's
political theory which seriously restricts the scope of its application. 7 He
argues that the theory can be made coherent only if Hobbes is assumed
to be speaking of a society which resembles our modern, bourgeois,
market societies. I believe that Hobbes's theory has a much greater range
of application than this. More specifically, I have argued that the
situations analysed by Hobbes are Prisoners' Dilemmas (possibly
iterated). These are neither identical with, nor are they only to be found
in, market societies. In this section, then, I must show how Macpherson's
argument fails.

There are two steps in the argument: (i) Macpherson claims that after
defining a man's 'power' as his means to obtain what he desires, Hobbes
proceeds to redefine power and that a 'new postulate is implied in this
redefinition of power, namely that the capacity of every man to get what
he wants is opposed by the capacity of every other man' (Macpherson, p.
36); (ii) 'the postulate that the power of every man is opposed to the
power of every other man requires the assumption of a model of society
which permits and requires the continual invasion of every man by every
other' (Macpherson, p. 42), and that the only such model of society is the
'possessive market society, which corresponds in essentials to modern
market society' (Macpherson, p. 68).

Each of these assertions is incorrect. Consider (i): first, Macpherson
believes that Hobbes speaks for the first time of the relations between
men, of man in society, only when he comes to discuss power. Yet
Hobbes has said earlier that all men desire eminence. Now clearly, desire
for eminence brings men into opposition with one another, for they
cannot all be eminent over others simultaneously. (And the greater the
ratio of 'eminence' to 'own payoff' in each man's utility function, the
more nearly the game approximates to a zero-sum game, or one of 'pure
opposition'.) Second, given that every man seeks to obtain what he
desires and given Hobbes's definition of power as the means to obtain

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what one desires, it follows (as we have seen) that every man desires
power; given further that man desires to be eminent, it follows that he
desires to have more power than others. No 'redefinition' of power, from
'absolute' to 'comparative' power, is involved here.

Macpherson seems to be aware that, if these two points are granted,
this first step in his argument is unnecessary (Macpherson, p. 45), and we
can pass immediately to the second and more important step.

If, as I have argued earlier, Hobbes's propositions about power
seeking can be derived from his definition of power and his proposition
that men seek to obtain the things they desire, so that 'power' plays no
logically essential role in Hobbes's political theory, then Macpherson's
assertion in the second part of his argument is clearly incorrect. But let us
see how he defends it.

Possessive market society is an ideal type to which modern capitalist
societies approximate. Its distinctive feature, as far as Macpherson's
argument is concerned, is that every individual owns his capacity to
labour and may sell it or otherwise transfer it as he wishes. The
consequence of this (and other assumptions) is that there is a market in
labour as well as in other commodities. It is this labour market which
provides the means by which 'the continual invasion of every man by
every other' is carried on. Labour markets may of course have this
property, but Macpherson is asserting that only societies with (amongst
other things) labour markets can provide such means. This is plainly
false, for there are many societies (and many more that have perished or
have been transformed) in which there is no market in labour (and in
some cases there are no markets in anything) and yet there is 'continual
invasion of every man by every other'. This 'invasion' may take several
forms. The primary objects of a man's desire may be the possession of
physical strength, skill in hunting, cattle and wives and good crops (if
there is individual ownership of these things), peace of mind, ceremonial
rank, and so on; his means to obtain these things, his 'power', may
include all of these things and others besides; and he may suffer
continual invasion and transfers of his power, simply because people
steal his women, cattle and foodstocks, hunt more skilfully, spread
rumours that he is a sorcerer, or whatever. None of this requires a market
in labour (or in anything else for that matter).

Macpherson's only defence against this would be to define power as
'access to the means of labour' or 'control of labour'. This would make

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the second part of his argument about possessive market society trivial.
At one point (p. 49) he seems to do just this, but then later (p. 56) he says
only that power must 'by definition include access to the means of
labour', which leaves room for power also to depend upon cattle and
ceremonial rank and all the rest.

I have argued that Leviathan is about Prisoners' Dilemmas, and this
means that Hobbes's argument, in the form in which I restated it, is not
confined to situations of the sort that Hobbes himself was obviously
most concerned about. If I am right, and if Prisoners' Dilemmas are to be
found outside possessive market societies, then Macpherson's argument
collapses. I believe that the problem Hobbes treats is to be found in one
form or another in most, if not all societies, including so-called primitive ^
societies with no markets in labour. In 'primitive' and other societies,
stealing one another's cattle, stealing corn from the communally owned
fields, or disturbing the tribe's tranquillity by excessive display, are
simple examples of behaviour which may lead to the problem Hobbes
was concerned with.

Although Macpherson's thesis is unacceptable, there is an interesting
proposition about possessive market society and the argument in
Leviathan, which I think has some plausibility. Very roughly, it is that the
more a society approximates to the possessive market type, the more
numerous are the sites and occasions for Prisoners' Dilemmas and the
greater is the severity of the Prisoners' Dilemmas, where by 'greater
severity' I mean a greater 'temptation' unilaterally to Defect from mutual
Cooperation. I could not, of course, begin to prove this.

I should add finally that while I disagree with Macpherson's view that
Hobbes's political theory is coherent only if society is assumed to be of
the possessive market variety, I nevertheless agree with him that Hobbes
seems to have been conscious of the possessive market nature of the
society in which he lived and that in Leviathan he sometimes speaks of
characteristic features of possessive market societies.

Hume's Leviathan

Hume's explanation of the necessity and desirability of government is
not very different from Hobbes's. But he begins with assumptions about
human nature which seem much less gloomy than those of Hobbes; his
explanation of the origin of government appears to be more plausible

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than Hobbes's contracterian account; and in place of the great
Leviathan that Hobbes sometimes made to sound so terrifying he
describes a government resembling the sort of governments that 'large
and civilis'd societies' in fact possess. Nevertheless, his assumptions
about human nature (which I shall discuss in the following section) are
effectively the same as those of Hobbes; his account of the origin of
government (pp. 159-60) rests on an analysis of the evolution of
property 'conventions' (pp. 154-5) which is itself not entirely plausible
(for reasons which I discuss on pp. 155-9); and as for the government
which Hume concludes to be necessary, its function is similar to that of
Hobbes's Leviathan and it must therefore be given as much power.
(Hence the title of this section.)

For all its essential similarity to Hobbes's theory, Hume's political
theory warrants a brief discussion here. First, because there are in fact
two new elements in Hume's account, which, though they have not been
given much attention by students of Hume, are important in the analysis
of voluntary cooperation and played an important role in my discussion
in chapters 3 and 4. Second, because it is Hume's version of the theory
rather than the stark account of Hobbes which was more acceptable to
later writers and to which many modern justifications of government
still largely correspond.

Individual preferences

I begin with a discussion of those elements of 'the passions' which are
incorporated in the assumptions on which Hume's political theory is
based.

(i) While Hobbes does not deny (in Leviathan at least) the existence in
some men of a positive altruism which is not reducible to egoism, he has
very little to say about it, and the effective assumption in his political
theory is that men's preferences reflect a combination of egoism and the
negative altruism which is involved in a desire for eminence. For Hume,
positive altruism, or 'benevolence', is more important. He distinguishes
two kinds of benevolence, 'private' and 'extensive'. Private benevolence
is a desire for the happiness of those we love, our family and friends. It is
not the same thing as love, but rather is a result of it; love is always
'follow'd by, or rather conjoin'd with benevolence . . .' (Tr 367). 8 This
private benevolence is an 'original instinct implanted in our nature', like

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love of life, resentment, kindness to children, hunger and 'lust' (Tr 368,
417, 439).

Extensive benevolence or 'pity' is 'a concern for . . . others, without
any friendship ... to occasion this concern or joy. We pity even
strangers, and such as are perfectly indifferent to us' (Tr 369). This kind
of benevolence is not instinctive; it is due to sympathy. Hume defines
'sympathy' with others as our propensity 'to receive by communication
their inclinations and sentiments, however different or even contrary to
our own' (Tr 316); it is 'the conversion of an idea into an impression by
the force of imagination' (Tr 427). This is not to say that sympathy is a
form of altruism. Nor is it to say, for example, that we suffer for ourselves
when we contemplate others suffering: we do not fear for our own lives
when we see, and sympathize with, others in danger of death and fearing
for their lives. Sympathy is simply the name for what makes it possible
for us to experience, to have an impression of, the feelings of others.

Sympathy, then, makes extensive benevolence possible. 'Tis true, there
is no human, and indeed no sensible, creature, whose happiness or
misery does not, in some measure, affect us, when brought near to us, and
represented in lively colours: . . . this proceeds merely from sympathy
. . .' (Tr 481). Again: 'We have no such extensive concern for society but
from sympathy' (Tr 579).

The important role played by sympathy in the Treatise is somewhat
reduced in the Enquiry. In particular, extensive benevolence, which was
due only to sympathy in the Treatise, now seems to be included with
private benevolence as one of the instincts. This view is given in the
Appendix on 'Self-Love' together with the argument (taken from Bishop
Butler's Fifteen Sermons, especially the first) that self-love is not our only
motivation - that there are 'instincts' (such as benevolence) which
motivate us directly and are not reducible to a species of self-love. It is
worth quoting Hume's argument at length:

There are bodily wants or appetites acknowledged by every one,
which necessarily precede all sensual enjoyment, and carry us directly
to seek possession of the object. Thus, hunger and thirst have eating
and drinking for their end ; and from the gratification of these primary
appetites arises a pleasure, which may become the object of another
species of desire or inclination that is secondary and interested. In the
same manner there are mental passions by which we are impelled

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immediately to seek particular objects, such as fame or power, or
vengeance without any regard to interest; and when these objects are
attained a pleasing enjoyment ensues, as the consequence of our
indulged affections. Nature must, by the internal frame and consti-
tution of the mind, give an original propensity to fame, ere we can
reap any pleasure from that acquisition, or pursue it from motives of
self-love, and desire of happiness. . . . Were there no appetite of any
kind antecedent to self-love, that propensity could scarcely ever exert
itself; because we should, in that case, have felt few and slender pains
or pleasures, and have little misery or happiness to avoid or to pursue.

Now where is the difficulty in conceiving, that this may likewise be
the case with benevolence and friendship, and that, from the original
frame of our temper, we may feel a desire of another's happiness or
good, which, by means of that affection, becomes our own good, and
is afterwards pursued, from the combined motives of benevolence and
self-enjoyments? (Enquiry, pp. 301-2.)

(ii) The operation of sympathy and the extent of benevolence are
limited by our manner of comparing ourselves with others. 'We seldom
judge of objects from their intrinsic value', says Hume, 'but form our
notions of them from a comparison with other objects; it follows that,
according as we observe a greater or less share of happiness or misery in
others, we must make an estimate of our own, and feel a consequent pain
or pleasure' (Tr 375). 'This kind of comparison is directly contrary to
sympathy in its operation . . .' (Tr 593), and accounts for the origin of
malice and envy (Tr 377). It is itself limited, inasmuch as men tend to
compare themselves with, and are envious of, only those who are similar
to them in relevant respects (Tr 377-8).

Negative altruism is real enough for Hume; but in his political theory
it does not assume the importance that it does in Hobbes's theory (as
part of the desire for eminence). Hume shrinks from making any general
statement, in the form of a simplifying assumption, about the predomin-
ance of positive or negative altruism. He allows that in some situations
positive altruism may dominate negative altruism, and vice versa in
other situations. But in the statement of his political theory, the effective
assumptions about individual preferences contain no reference to
negative altruism; as we shall see shortly, they refer only to egoism,
limited positive altruism and 'shortsightedness'.

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(iii) Although Hume argues for the existence of an independent
motive of private benevolence and that extensive benevolence or pity is
found in all men, since they are all capable of sympathy (Tr 317, 481),
nevertheless it is clear that, when he comes to explaining the origins of
justice, property and government, he assumes that benevolence is very
limited. In one place, he suggests that each individual loves himself more
than any other single person, but the aggregate of his benevolent
concerns for all others exceeds his self-love (Tr 487). But more generally,
he says that some men are concerned only for themselves, and that, as for
the others, their benevolence extends only or chiefly to their family and
friends, with only a very weak concern for strangers and indifferent
persons (Tr 481, 489, 534).

Hume is not very precise about the relative weights of benevolence
and self-interest. All we can say is that, in his political theory, his
assumption is effectively that men are self-interested and benevolent, but
that the benevolence is not so great that there is no need for 'conventions'
about property (Tr 486, 492, 494-5). These will be explained below.

(iv) There is another element in the structure of individual pre-
ferences, to which Hume (in the Treatise) attaches great importance : we
discount future benefits, their present value to us diminishing as the
future time at which we expect to receive them recedes farther from the
present. What is close to us in time or space, says Hume, affects our
imagination with greater force than what is remote, the effect of time
being greater than that of space (Tr 427-9). The consequence of this is
that men 'are always much inclin'd to prefer present interest to distant
and remote; nor is it easy for them to resist the temptation of any
advantage, that they may immediately enjoy, in apprehension of an evil,
that lies at a distance from them' (Tr 539, 535).

Property

Hume distinguishes 'three different species of goods' which we may
possess : mental satisfactions, our natural bodily endowments, and 'such
possessions as we have acquir'd by our industry and good fortune'. Only
the third species, external possessions, may be transferred unaltered to
others and used by them (Tr 487-8).

These external possessions are the source of 'the principal disturbance
in society' and this is because (i) they are scarce and easily transferred

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between people (Tr 488-9); (ii) everyone wants them: 'This avidity
alone, of acquiring goods and possessions for ourselves and our nearest
friends, is insatiable, perpetual, universal, and directly destructive of
society. There scarce is any one, who is not actuated by it ; and there is no
one, who has not reason to fear from it, when it acts without any restraint
. . .' (Tr 491-2); and (iii) man's selfishness in the pursuit of them is
insufficiently counteracted by his benevolence towards others to make
him abstain from their possessions (Tr 492, 486-8).

The resulting situation is essentially the same as Hobbes's 'state of
nature', though Hume has described it in less dramatic terms. The only
remedy for it is a 'convention enter'd into by all the members of the
society to bestow stability on the possession of those external goods, and
leave every one in the peaceable enjoyment of what he may acquire by his
fortune and industry' (Tr 489). However, a permanent 'stability' of
possession would itself be 'a grand inconvenience', for 'mutual exchange
and commerce' is necessary and desirable. Therefore there must also be a
'convention' facilitating the transfer of possessions by consent (Tr 514).
This in turn would be of little use without a 'convention' to keep one's
promises, since it is usually impracticable for the parties to an exchange
to transfer possessions simultaneously (Tr 516-22).

There are thus three 'conventions' which men must make to obtain
'peace and security' : 'that of the stability of possession, of its transference
by consent, and of the performance of promises'. These are the 'laws of
justice' or 'the three fundamental laws of nature' (Tr 526). 'Property' can
now be defined as 'nothing but those goods, whose constant possession
is establish'd ... by the laws of justice' (Tr 491); and we can say that
'justice' consists in the observation of the current laws fixing the
distribution of property and protecting the parties to exchanges of
property.

Conventions

A convention, says Hume, is not like a promise; for promises themselves
arise from human conventions (Tr 490). Conventions, he means to tell
us, are not like the covenants which, according to Hobbes, are the only
means of escape from the state of nature. A convention is rather

a general sense of common interest; which sense all the members of
the society express to one another, and which induces them to

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regulate their conduct by certain rules. I observe, that it will be for my
interest to leave another in the possession of his goods, provided he
will act in the same manner with regard to me. He is sensible of a like
interest in the regulation of his conduct. When this common sense of
interest is mutually express'd, and is known to both, it produces a
suitable resolution and behaviour. And this may properly enough be
call'd a convention or agreement betwixt us, tho' without the
interposition of a promise; since the actions of each of us have a
reference to those of the other, and are perform'd upon the
supposition, that something is to be perform'd on the other part. (TV
490)

Now this is a perfectly reasonable definition of convention; it is
roughly what we still typically mean by convention. But then, it seems to
me, the laws of justice are not conventions. If they were, there would be
no need for a government to constrain people to conform to them, as
Hume goes on to argue.

Since this point is rather important, it is worth giving here a more
precise definition of convention. We can use the one constructed by
David Lewis in his Convention : A Philosophical Study.

Conventions are solutions to coordination problems. The most clear-
cut case of a coordination problem (to which we may confine our
attention) is the situation facing the players in a game of pure
coordination. This is a game, having two or more proper coordination
equilibria, and in which the players' interests coincide, so that their
payoffs at each outcome are equal. A coordination equilibrium is a
strategy vector such that no player can obtain a larger payoff if he or any
other player unilaterally uses a different strategy (so that a coordination
equilibrium is an equilibrium, as denned in chapter 3, but not
conversely); and a coordination equilibrium is proper if each player
strictly prefers it to any other outcome he could obtain, given the other
strategy choices. Thus the two-person game with the payoffs shown in
Matrix 1 below is a pure coordination game; strategy vectors (r l5 c t ) and
(r 2 , c 2 ) are proper coordination equilibria, while (r 3 , c 3 ) is improper.

c 3

r l

2,2

0,0

r 2

0,0

2,2

0,0

1, 1

(Matrix 1)

THE STATE 157

A simple example of a coordination problem is the situation facing
two people who are not in communication and who wish to meet but are
indifferent between several alternative meeting-plates. Suppose there are
just three possible meeting-places. Then the payoff matrix is that shown
as Matrix 2 (the payoffs there being merely ordinal).

1, 1

0,0

1, 1

0,0

(Matrix 2)

0,0

1, 1

Another simple example is that of several drivers on the same road ;
nobody cares which side of the road he drives on, as long as everybody
else drives on the same side as he does. This is an example of an iterated
or recurrent coordination problem.

The definition of a coordination problem requires that there be at least
two coordination equilibria. If there is only one, the problem is trivial, for
the players will have no difficulty in coordinating their choices.

We are now in a position to define convention.

A regularity R in the behaviour of members of a population P when
they are agents in a recurrent situation S is a convention if and only if,
in any instance of S among members of P,

(1) everyone conforms to R;

(2) everyone expects everyone else to conform to R ;

(3) everyone prefers to conform to on condition that the
others do, since S is a coordination problem and uniform
conformity to A is a coordination equilibrium in S. 9

Players in a coordination game will achieve coordination if they have
what Lewis calls 'suitably concordant mutual expectations'. If a player is
sufficiently confident that the others will do their parts of a particular
coordination equilibrium, then he will do his part. Where communi-
cation is possible, agreement is the simplest means of producing
concordant mutual expectations and hence coordination, but a conven-
tion need not be started by an agreement. In a recurrent coordination
problem, concordant mutual expectations may be built up gradually, as
more and more people conform to a regularity, until a convention is
established. Thus, without an explicit agreement and without any
coercion, a convention to drive on a particular side of the road could be
expected to grow up: each man prefers to drive on the side of the road on
which most others are driving; at some stage of the process, more or less

158

THE STATE

159

by chance, a majority will be driving on the left, say; this produces or
strengthens an expectation in each driver that a majority will in the
future drive on the left; and in this way, a convention to drive on the left
is very quickly established.

Lewis's definition of convention is (as he himself recognizes) essen-
tially the same as the one given by Hume. Hume, too, recognizes that \
conventions will emerge 'spontaneously', without agreements or govern- \
ments. Speaking of the conventions on property, he says that when a {
'common sense of interest is mutually express'd, and is known to both, it j
produces a suitable resolution and behaviour'; and: 'Nor is the rule
concerning the stability of possession the less deriv'd from human |
conventions that it arises gradually, and acquires force by a slow
progression, and by our repeated experiences of the inconveniences of
transgressing it. On the contrary, this experience assures us still more,
that the sense of interest has become common to all our fellows, and
gives us a confidence of the future regularity of their conduct : And 'tis
only on the expectation of this, that our moderation and abstinence are
founded' (Tr 490).

Conventions not only emerge but also persist spontaneously; for a f
convention is an equilibrium, from which no individual has an incentive f
unilaterally to deviate. It follows that everyone will conform to a ■'■
convention without being coerced by a government or by any other
agency. 10 Yet Hume goes on to argue that men will not voluntarily
observe the conventions they make about property and government is
necessary to constrain them to conform. The reason he gives for this, as I
shall argue in the next part, is essentially that men find themselves, not in
a recurrent coordination game, but in a recurrent or iterated Prisoners'
Dilemma game (with future payoffs discounted). If this is the case, then
the laws of justice cannot be conventions. And for precisely the same
reason that men will not voluntarily observe their property conventions,
these conventions would not emerge spontaneously in the first place.

It would be proper to call the laws of justice 'conventions' only if all
men preferred any system of such laws (and therefore any distribution of
possessions) to no laws at all and were indifferent (or nearly so) between
all possible systems. The first condition is accepted by Hume, for, like
Hobbes, he believes that 'without justice, society must immediately
dissolve and fall into that savage and solitary condition, which is
infinitely worse than can possibly be suppos'd in society', so that, upon

the introduction of the laws of justice 'every individual person must find
himself a gainer . . .' (Tr 497). As for the second condition, it is true that
in the Enquiry Hume remarks that 'What possessions are assigned to
particular persons; this is, generally speaking, pretty indifferent . . .'
(Enquiry, p. 309 note). But this remark is quite contrary to the
assumption, which is essential to his whole theory, that men have an
'insatiable, perpetual, universal' avidity for acquiring external posses-
sions. Men are certainly not indifferent between different distributions of
property and therefore are not indifferent between different laws of
justice, which determine the distributions.

The necessity of government

According to Hume, government is necessary in large societies because
without it men will not observe the laws of justice; and it is on the
observance of these laws alone that 'the peace and security of human
society entirely depend' (Tr 526; see also 491). His argument that men
will not keep the laws of justice in large societies has two threads. The
first is essentially the argument given by Olson in The Logic of Collective
Action, which we considered in chapter 1. The second concerns the
discounting of future benefits, which played such an important role in
chapters 3 and 4. Hume does not maintain a clear distinction between
these two elements. Nevertheless, the spirit of this part of his theory is
that men will not voluntarily cooperate (abstain from each other's
possessions; observe the laws of justice) because they are players in a
Prisoners' Dilemma supergame and their discount rates are too great.
The 'size' argument appears clearly in the following passage :

Two neighbours may agree to drain a meadow, which they possess in
common; because 'tis easy for them to know each others mind; and
each must perceive, that the immediate consequence of his failing in
his part, is the abandoning the whole project. But 'tis very difficult,
and indeed impossible, that a thousand persons shou'd agree in any
such action; it being difficult for them to concert so complicated a
design, and still more difficult for them to execute it ; while each seeks
a pretext to free himself of the trouble and expence, and wou'd lay the
whole burden on others. (Tr 538)

Hume gives here both of the reasons why, according to Olson, large
groups do not provide themselves with public goods, such as a drained

160

THE STATE

meadow shared by the group: first, each individual member has no
incentive to make his contribution because it is a public good which is
being provided and he therefore benefits from it, if it is provided at all,
whether he contributes or not; second, and less important, the larger the
group the greater are the costs of organization.

Hume makes it quite clear that this part of his argument applies only
to large societies, and several times proclaims his belief that the members
of small societies may voluntarily conform to the property 'conventions'
and may therefore live without government (Tr 499, 539-41, 543, 546,
553-4). But this is partly because small societies tend to be 'uncultivated',
that is, they do not have very many possessions to quarrel about.

In the meadow-drainage example which I have quoted from the
Treatise, Hume deals only with the 'static' part of his argument. But
elsewhere, whenever he presents the 'logic of collective action', it is
bound up with the proposition (which I discussed earlier) that men
discount future benefits. Men '. . . prefer any trivial advantage, that is
present, to the maintenance of order in society, which so much depends
on the observance of justice. The consequences of every breach of equity
seem to lie very remote, and are not able to counterbalance any
immediate advantage, that may be reap'd from it' (Tr 535; see also 499,
537-9, 545).

In the continuation of this passage, Hume in effect speaks of behaviour
in a sequence of Prisoners' Dilemmas: when you commit acts of injustice
as well as me, 'Your example both pushes me forward in this way by
imitation, and also affords me a new reason for any breach of equity, by
shewing me, that I should be the cully of my integrity, if I alone should
impose on myself a severe restraint amidst the licentiousness of others'
(Tr 535).

The only remedy for this situation is to establish government. The
only way men can obtain security and peace is to induce a few men,
'whom we call civil magistrates, kings and their ministers', to constrain
every member of the society to observe the laws of justice (Tr 537).

Thus, Hume's case for government rests on the alleged inability of men
to cooperate voluntarily in the provision of peace and security. However,
he goes on to add that 'government extends farther its beneficial
influence' by forcing men to cooperate in the provision of other public
goods. Thus, he says, 'bridges are built; harbours open'd; ramparts
rais'd; canals form'd; fleets equipp'd; and armies disciplin'd; every
where, by the care of government . . .' (Tr 538-9).

THE STATE

161

Hume and Hobbes

The assumptions about the structure of static individual preferences on
which Hume bases his political theory are not quite the same as those
made by Hobbes. To use the language of chapter 5, Hobbes assumes that
each man's preferences are a combination of egoism and negative
altruism, reflecting a desire to maximize his own payoff and his
eminence, whereas Hume assumes that they are a combination of egoism
and positive altruism, with egoism predominant. However, the effect is
the same in both cases: the resulting game at any point in time is a
Prisoners' Dilemma. If in both cases the payoffs are such that the game is
a Prisoners' Dilemma when only pure egoism is assumed on the part of
each player, then we can say that the 'transformed game' (the game
which results when altruism is introduced) is a more severe Prisoners'
Dilemma under Hobbes's assumptions than under Hume's.

This assumption of Hume's about preferences applies only to men in
large societies. Hume is aware that in sufficiently small societies the game
may not be a Prisoners' Dilemma, and here he largely anticipates the
ideas which form the core of Olson's argument. Hobbes, on the other
hand, does not discuss these ideas; but we cannot say that he was
unaware of them, for in Leviathan he apparently has in mind only large
societies (especially the one in which he lived) and accordingly assumes
in effect that the game is a Prisoners' Dilemma.

There is another important element in Hume's argument which is
largely absent from Hobbes's, namely time. I have already commented
on the fact that, although Hobbes's argument is not entirely static, there
is no reference to intertemporal preferences in his assumptions; no
account is taken of the discounting of future benefits, which, as we saw in
chapters 3 and 4, plays such a crucial role in determining whether
voluntary Cooperation will occur in sequences of Prisoners' Dilemma
games. Hume's treatment is in this respect more realistic than Hobbes's.
Time appears in his assumptions about individual preferences: future
payoffs are to be discounted in calculating their present value. This fact
plays an important role in his argument, for the discount rate is a
principal reason why men do not voluntarily cooperate in observing the
laws of justice.

Hume is not so specific in his detailed assumptions and arguments
that one can make precise comparisons of his theory with that of Hobbes
or with the analysis of the Prisoners' Dilemma supergame given in

162

THE STATE

chapters 3 and 4. We certainly cannot say, for example, that Hume
understood (what is shown in chapter 4) that Cooperation is rational
throughout an JV-person Prisoners' Dilemma supergame only if the
players adopt conditional strategies of a certain form and a certain
inequality relating the discount rate and the payoff functions is satisfied
for each player. We cannot even say that the Treatise contains an
analysis of the Prisoners' Dilemma supergame. Nevertheless, the general
outline of Hume's theory is quite clear and we can say that there is an
approximate similarity between his ideas and parts of the analysis in
chapter 3. If this comparison is legitimate, then we can say that Hume
failed to appreciate that even when the society is so large that the
ordinary game is a Prisoners' Dilemma, Cooperation in the supergame
may yet be rational if the individual discount rates are not too great.

Despite his more 'dynamic' treatment of the problem, Hume comes to
essentially the same conclusion as Hobbes: governments, powerful
enough to enforce 'justice' and maintain Peace, are necessary and
desirable. The comment made earlier on Hobbes's conclusion applies to
Hume also: from their assumptions (including, in Hume's case, the
assumption of a 'high' discount rate), one can deduce only that some
form of coercion is necessary to establish or maintain Peace; one cannot,
strictly speaking, conclude that this coercion must take the form of
government.

My final comment on Hobbes and Hume, before I turn in the next
chapter to consider more fundamental criticisms of their approach,
concerns the assumption, which is absolutely essential to their argu-
ments, that 'the greatest, that in any forme of Government can possibly
happen to the people in generall, is scarce sensible, in respect of the
miseries, and horrible calamities, that accompany a Civill Warre; or that
dissolute condition of masterlesse men . . .' (Lev 141; for an almost
identical statement by Hume, see Tr 497). In other words, it is assumed
that government-enforced Peace is preferred by every individual to the
state of War no matter how great are the costs of government.

Now the only kinds of costs which Hobbes and Hume appear to have
in mind in this connection are those which are to be merely subtracted, so
to speak, from the benefits of mutual Cooperation (the resulting utility
for the mutual Cooperation outcome being for every individual
diminished but still greater than that of the mutual non-Cooperation
outcome). Yet a government powerful enough to enforce Cooperation

THE STATE

163

may impose costs of other kinds. In the first place, it may diminish the
desirability of the state of Peace per se (in addition, that is, to imposing
costs merely in order to ensure this outcome). This is because people
tend to derive more satisfaction from doing things which are initiated
and carried out spontaneously and voluntarily than from doing the same
things at the suggestion and command of others, including the state.

Secondly, the state may have cumulative effects on the very conditions
which, according to Hobbes and Hume, make states necessary. In
particular, it may over a period of time cause a Prisoners' Dilemma to
appear where none existed before, or cause an already existing Prisoners'
Dilemma to become more severe. Dynamical effects of this sort are of a
wholly different order from those mentioned earlier, and I believe that
the entire approach to the justification of the state which has been
considered in this chapter is undermined if they are taken seriously. I
shall try to take them seriously in the next chapter.

7. Epilogue: cooperation, the state and
anarchy

By his entry into any society the individual . . . offers up a portion of (his)
liberty so that society will vouchsafe him the rest. Anybody who asks for an
explanation is usually presented with a further saying: 'The liberty of each
human being should have no limits other than that of every other.' At first glance,
this seems utterly fair, does it not? And yet this theory holds the germ of the
whole theory of despotism. 1

Bakunin, L'Empire Knouto-Germanique

Therefore we can only repeat what we have so often said concerning authority
in general: 'To avoid a possible evil you have recourse to means which in
themselves are a greater evil, and become the source of those same abuses that
you wish to remedy . . .'

Kropotkin, The Conquest of Bread !

■j

The treatment of the problem of voluntary cooperation in the first four
chapters and the political theories of Hobbes and Hume as I presented
them in chapter 6 rest solely on assumptions about individuals. These
assumptions embody a conception of the individual as being endowed
with a given and unchanging structure of preferences. More specifically, it |
is assumed that each individual is characterized by a certain combi-
nation of egoism and some form of altruism, and it is further assumed
that this characterization does not change with time. His preferences are
treated as exogenous to what has to be explained (or justified) by the |
theories in question. They are independent of, and do not change in
response to, his social situation. He is an example of what Marx called
the 'abstract man'.

This means, in particular, that no account is taken of the effect on
individual preferences of the activities of the state or of the activities of *
the individuals themselves. If the activities of the state may result in
changes in individual preferences, then clearly it cannot be deduced from

epilogue: cooperation, the state and anarchy 165

the structure of preferences in the absence of the state that the state is
desirable. More generally, if individual preferences change (not necess-
arily as a result of state activity), the question of the desirability (or
'preferability') of the state becomes much more complex than it is in the
static theories we have been considering; and if preferences change as a
result of the state itself, then it is not even clear what is meant by the
desirability of the state.

The effects of the state on individual preferences and the ways in which
preferences may change in the absence of the state are the subjects of the
main section of this final chapter (The decay of voluntary cooperation').
I shall suggest there (rather inconclusively, it has to be admitted) that the
effect of the state is to exacerbate the very conditions which are claimed
to provide its justification and for which it is supposed to provide a
partial remedy. In two preliminary sections I shall mention - much less
controversially, it seems to me - two other ways in which states create or
aggravate problems of the kind they are supposed to solve and
undermine conditions for alternatives to the state to be workable.

In what follows I take the state to be (amongst other things) a complex
system of interacting, partially independent components (like: police,
security and military forces, an executive, legislature, judiciary, ad-
ministrative service, and so on), and when I speak of the effects of state
action I shall be referring to the aggregate (or outcome or resultant) of
the components' actions which are in turn the aggregate of the actions of
the individuals who staff them.

International anarchy

If, as the liberal theory argues, a state is an effective way of solving the
two fundamental collective action problems of maintaining order
internally and providing defence against external enemies and competi-
tors, then the very process of becoming politically more centralised - of
building or strengthening a state - is likely to be seen as threatening by
neighbouring and competitor societies, and the response is likely to be
the formation or strengthening of their own states, and in particular, of
course, the building up of their own 'defensive' capabilities. The structure
of preferences involved in this process, which can characterize relations
between societies of any sort, not merely those which are usually called
nations, is likely to be that of a Prisoners' Dilemma game. Interactions of

164

166 epilogue: cooperation, the state and anarchy

this kind may also, however, generate other collective action problems,
including some representable as Chicken games. 2

If this is so, then we could say that states, established at one level (the
national level, for example) to rescue people in a ('domestic') Prisoners'
Dilemma or other collective action problem, may cause a Prisoners'
Dilemma or other collective action problem to emerge at another level
(the international level, for example) or exacerbate an already existing
one.

Hobbes himself noted that 'Sovereigns', who alone can save people
from the state of (domestic) 'War', are themselves in a 'state of nature',
without a 'common power to keep them all in awe' :

But though there had never been any time, wherein particular men
were in a condition of warre one against another; yet in all times,
Kings, and Persons of Sovereigne authority, because of their
Independency, are in continuall jealousies, and in the state and
posture of Gladiators; having their weapons pointing, and their eyes
fixed upon one another; that is, their Forts, Garrisons, and Guns
upon the Frontiers of their Kingdoms; and continuall Spyes upon
their neighbours; which is a posture of War. (Lev 98)

Nevertheless, neither Hobbes nor Hume applied to the international
'state of nature' the analysis which they made of the domestic one. But
there is no reason in principle why such an application should not be
made. Many people have of course done just this, arguing that a
supranational state is necessary if international collective action prob-
lems, including that of the maintenance of international peace, are to be
solved. And contrariwise, the possibility of conditional cooperation
amongst states in the absence of such a supranational state has been
taken more seriously in the last few years. 3

The destruction of community

Hume argues that in large societies life without government is appalling,
but that in small societies this need not be the case. Therefore, he says,
people in a large society need, and will in fact establish, a government.
When the argument is put this way, however, a radically different
conclusion suggests itself: that large societies should be (or will be)
disaggregated into smaller societies, and the enlargement of societies and

epilogue: cooperation, the state and anarchy 167

the destruction of small ones should be (or will be) resisted. This
conclusion does not follow logically from Hume's premises any more
than does his own conclusion. Given these premises (or those of
Hobbes), the most that we can assert in this connection is that the larger
the society, the less likely it is that there will be voluntary cooperation in
the provision of public goods and in the solution of other collective
action problems, principally because of the increased difficulty of
conditional cooperation. If the relations between the members of a
smaller group are those characteristic of community, then the usual
range of positive and negative sanctions, including informal social
sanctions, that are most effective in small communities, can also help to
maintain cooperation in the absence of the state (though it should not
then be called 'voluntary'), both directly and (like the state in Hobbes's
account) indirectly through bolstering conditional cooperation.

In view of this, it is perhaps ironical that the state should be presented
as the saviour of people caught in the Prisoners' Dilemmas (and other
collective action problems) of a large society; for historically the state
has undoubtedly played a large part in providing the conditions in which
societies could grow and indeed in systematically building large societies
and destroying small communities. The state has in this way acted so as
to make itself even more necessary.

Of course, states were not alone in causing the decline of community
and it is difficult to disentangle their contributions from those of other
causes such as the expansion of industrial capitalism ; but that the state
had an important independent effect there can be no doubt.

I am not thinking so much of the very origin of the long process of state
formation, when the normal process of fissioning that is characteristic of
stateless societies is inhibited. Such fissioning, whereby a part of a
community breaks away and establishes a replica community elsewhere,
ensures that the society continues to be composed of small communities.
When this is no longer possible, communities must grow in size or
become joined to others. This is part of the process that leads to the
emergence of a state. But what I have more in mind is the 'self-building' of
states through the intentional destruction or absorption or weakening of
(small) communities and the concomitant construction or extension or
strengthening of nations or other larger societies, which can only be
communities in a much weaker sense. 4 This is as true of the growth of the
earliest states and of the modern European states as it is of many nations

168 epilogue: cooperation, the state and anarchy

made independent since the Second World War, where the new states
have often quite consciously set about weakening loyalty to ethnic and
other groups within the proto-nation in order to build a single 'national
solidarity'. 5

The state, then, has in this way tended to exacerbate the conditions
which are claimed (in the liberal theory) to provide its justification and
for which it is supposed to be the remedy. It has undermined the
conditions which make the principal alternative to it workable and in
this way has made itself more desirable.

The decay of voluntary cooperation \

The arguments for the necessity of the state which I am criticizing in this I
book are founded on the supposed inability of individuals to cooperate f
voluntarily to provide themselves with public goods, and especially, in I
the theories of Hobbes and Hume, with security of person and property, f
The intervention of the state is necessary, according to these arguments, j
in order to secure for the people a Pareto-optimal provision of public f
goods, or at least to ensure that some provision is made of the most t
important public goods.

In this section I suggest that the more the state intervenes in such
situations, the more 'necessary' (on this view) it becomes, because
positive altruism and voluntary cooperative behaviour atrophy in the '•
presence of the state and grow in its absence. Thus, again, the state
exacerbates the conditions which are supposed to make it necessary. We
might say that the state is like an addictive drug : the more of it we have,
the more we 'need' it and the more we come to 'depend' on it. >

Men who live for long under government and its bureaucracy, courts J
and police, come to rely upon them. They find it easier (and in some cases
are legally bound) to use the state for the settlement of their disputes and
for the provision of public goods, instead of arranging these things for ,
themselves, even where the disputes, and the publics for which the goods
are to be provided, are quite local. In this way, the state mediates between |
individuals; they come to deal with each other through the courts,
through the tax collector and the bureaucracies which spend the taxes. In
the presence of a strong state, the individual may cease to care for, or
even think about, those in his community who need help; he may cease
to have any desire to make a direct contribution to the resolution of local

epilogue: cooperation, the state and anarchy 169

problems, whether or not he is affected by them ; he may come to feel that
his responsibility to society has been discharged as soon as he has paid
his taxes (which are taken coercively from him by the state), for these
taxes will be used by the state to care for the old, sick and unemployed, to
keep his streets clean, to maintain order, to provide and maintain
schools, libraries, parks, and so on. The state releases the individual from
the responsibility or need to cooperate with others directly ; it guarantees
him a secure environment in which he may safely pursue his private
goals, unhampered by all those collective concerns which it is supposed
to take care of itself. This is a part of what Marx meant when he wrote (in
'On the Jewish Question') of state-enforced security as 'the assurance of
egoism'.

The effects of government on altruism and voluntary cooperation can
be seen as part of the general process of the destruction of small societies
by the state which was described earlier. The state, as we have seen,
weakens local communities in favour of the larger national society. In
doing so, it relieves individuals of the necessity to cooperate voluntarily
amongst themselves on a local basis, making them more dependent upon
the state. The result is that altruism and cooperative behaviour
gradually decay. The state is thereby strengthened and made more
effective in its work of weakening the local community. Kropotkin has
described this process in his Mutual Aid. All over Europe, in a period of
three centuries beginning in the late fifteenth century, states or proto-
states 'systematically weeded out' from village and city all the 'mutual-
aid institutions', and the result, says Kropotkin, was that

The State alone . . . must take care of matters of general interest, while
the subjects must represent loose aggregations of individuals, con-
nected by no particular bonds, bound to appeal to the Government
each time that they feel a common need.

The absorption of all social functions by the State necessarily
favoured the development of an unbridled, narrowminded individu-
alism. In proportion as the obligations towards the State grew in
numbers the citizens were evidently relieved from their obligations
towards each other. 6

Under the state, there is no practice of cooperation and no growth of a
sense of the interdependence on which cooperation depends; there are
fewer opportunities for the spontaneous expression of direct altruism

170 epilogue: cooperation, the state and anarchy

and there are therefore fewer altruistic acts to be observed, with the result
that there is no growth of the feeling of assurance that others around one
are altruistic or at least willing to behave cooperatively - an assurance
that one will not be let down if one tries unilaterally to cooperate.

A part of this argument has recently been made by Richard Sennett.
Sennett's interest is in reversing the trend towards 'purified' urban and
suburban communities through the creation of cities in which people
would learn to cope with diversity and 'disorder' through the necessity of
having to deal with each other directly rather than relying on the police
and courts and bureaucracies. The problem, he says, is 'how to plug
people into each others' lives without making everyone feel the same'.
This will not be achieved by merely devolving the city government's
power onto local groups :

Really decentralized power, so that the individual has to deal with
those around him, in a milieu of diversity, involves a change in the
essence of communal control, that is, in the refusal to regulate conflict.
For example, police control of much civil disorder ought to be sharply
curbed ; the responsibility for making peace in neighbourhood affairs
ought to fall to the people involved. Because men are now so innocent
and unskilled in the expression of conflict, they can only view these
disorders as spiralling into violence. Until they learn through
experience that the handling of conflict is something that cannot be
passed on to the police, this polarization and escalation of conflict
into violence will be the only end they can frame for themselves. 7

In his remarkable study of blood donorship, The Gift Relationship,
Richard Titmuss has given us an example of how altruism generates
altruism - of how a man is more likely to be altruistic if he experiences or
observes the altruism of others or if he is aware that the community
depends (for the provision of some public good) on altruistic acts. 8 The
availability of blood for transfusion is of course a public good. In
England and Wales, all donations are purely voluntary (with the partial
exception of a very small amount collected under pressure from prison
inmates). In the United States, only 9 per cent of donations were purely
voluntary in 1967 (and the percentage was falling). Of the rest, most are
paid for or are given 'contractually' (to replace blood received instead of
paying for it, or as a 'premium' in a family blood insurance scheme). As
Titmuss recognizes, even the donors he calls 'voluntary' (those who do

epilogue: cooperation, the state and anarchy 171

not receive payment, do not give contractually, and are not threatened
directly with tangible sanctions or promised tangible rewards) must
have 'some sense of obligation, approval and interest'. Nevertheless, the
voluntary donation of blood does seem to approximate as closely as is
perhaps possible to the ideal of pure, spontaneous altruism : for it is given
impersonally and sometimes with discomfort, without expectation of
gratitude, reward or reciprocation (for the recipient is usually not known
to the donor), and without imposing an obligation on the recipient or
anyone else; and 'there are no personal, predictable penalities for not
giving; no socially enforced sanctions of remorse, shame or guilt'. 9 It is,
then, an example of the kind of altruism which Hume specifically
declared to be very limited or absent; it is precisely not the 'private
benevolence' towards family and friends which he thought was common.

Now, if there is any truth in the general argument about the growth
and decay of altruism which was put forward above, we should at least
expect that the growth of voluntary donations should be greater in a
country in which non-voluntary donations are absent than in one where
they are present, and even that voluntary donations should decline with
time in a country where a very large proportion of donors were non-
voluntary. This is precisely what has happened in the countries which
Titmuss examines. In the developed countries the demand for blood has
risen very steeply in recent years, much more steeply than the
population. Yet in England and Wales, from 1948 to 1968, supply has
kept pace with demand, and there have never been serious shortages. On
the other hand, in the United States, in the period 1961-7 for which
figures are available, supply has not kept pace with demand and there
have been serious shortages; even more significantly, those blood banks
which paid more than half of their suppliers collected an increasing
quantity of blood in this period, while the supply to other banks
decreased. In Japan, where the proportion of blood which is bought and
sold has risen since 1951 from zero to the present 98 per cent, shortages
are even more severe than in the United States.

These differences, between England and Wales on the one hand and
America and Japan on the other, are consistent with the hypothesis that
altruism fosters altruism (though of course they do not confirm it).
Support (also inconclusive) for this explanation of the growth of blood
donations in England comes from some of the responses to a question
included in Titmuss's 1967 survey of blood donors in England: 'Could

172 epilogue: cooperation, the state and anarchy

you say why you first decided to become a blood donor?'. Many people,
it appears, became blood donors as a result of experiencing altruism:
they or their friends or relatives had received transfusions. For example:

To try and repay in some small way some unknown person whose
blood helped me recover from two operations and enable me to be
with my family, that's why I bring them along also as they become old
enough. (Married woman, age 44, three children, farmer's wife)

'Some unknown person gave blood to save my wife's life. (Married '
man, age 43, two children, self-employed windowcleaner)

Some responses hint at an altruism resulting from an appreciation of the
dependence of the system on altruism and of people's dependence on each
other:

You can't get blood from supermarkets and chaine stores. People
themselves must come forward, sick people cant get out of bed to ask
you for a pint to save thier life so I came forward in hope to help
somebody who needs blood. (Married woman, aged 23, machine
operator) 10

Peter Singer, in his discussion of Titmuss's book, has drawn attention
to some experiments which also support the hypothesis that altruism is
encouraged by the observation of altruism. 11 He mentions an experi-
ment in which a car with a flat tyre was parked at the side of the road with
a helpless-looking woman standing beside it. Drivers who had just
passed a woman in a similar plight but with a man who had stopped to
change her wheel for her (this scene having of course been arranged by
the experimenters) were significantly more likely to help than those who
had not witnessed this altruistic behaviour. 12 Singer himself writes : 'I
find it hardest to act with consideration for others when the norm in the
circle of people I move in is to act egotistically. When altruism is
expected of me, however, I find it much easier to be genuinely altruistic'

The argument I have made in this section is not of course new. A
similar (though not identical) argument is familiar to us from the
writings of the classical liberals, and especially of John Stuart Mill. With
the partial exception of Kropotkin, the only anarchist writer who makes
full and explicit use of something like this argument is William Godwin.
(Though Godwin is not wholly an anarchist. His case against govern-
ment in the Enquiry Concerning Political Justice represents in most

epilogue: cooperation, the state and anarchy 173

respects a more extensive and more throughgoing application of Mill's
argument than Mill himself makes.)

For Godwin, government is an evil which is necessary only as long as
people behave in the way in which they have come to behave as a result of
living for a long time under government. If governments were dissolved,
he says 'arguments and addresses' would not at first suffice to persuade
people to 'cooperate for the common advantage' and 'some degree of
authority and violence would be necessary. But this necessity does not
appear to arise out of the nature of man, but out of the institutions by
which he has been corrupted.' 13 Later, government would not be
necessary at all: there would be a transition to anarchy during which
people would learn to cooperate voluntarily (or, at least, to cooperate in
order to avoid the disapprobation of neighbours: 'a species of coercion'
which would presumably be effective in the small 'parishes' of Godwin's
ideal social order 14 ). The growth of cooperation would in part result
from the growth of benevolence. Benevolence is 'a resource which is
never exhausted' but becomes stronger the more it is exercised; and if
there is no opportunity for its exercise, it decays. The idea permeates
much of Godwin's Enquiry; we see it, for example, in his criticism of
punishment by imprisonment:

Shall we be most effectually formed to justice, benevolence and
prudence in our intercourse with each other, in a state of solitude?
Will not our selfish and unsocial dispositions be perpetually in-
creased? What temptation has he to think of benevolence or justice,
who has no opportunity to exercise it? 15

At the same time as Godwin wrote the Enquiry Concerning Political
Justice, Wilhelm von Humboldt was composing The Limits of State
Action, a book which contains many of the ideas to be found in the
Enquiry, especially those which are of interest here. 16 Humboldt was
certainly not an anarchist; but he did argue that the scope of state
activity should be strictly limited to the provision of 'mutual security and
protection against foreign enemies', and his case against the further
interference of the state rested on arguments similar to Godwin's and
more fundamentally on the axiom (on which Mill's On Liberty was also
to be based) that '. . . the chief point to be kept in view by the State is the
development of the powers of its citizens in their full individuality.' 17

By security, Humboldt meant 'the assurance of legal freedom':

174 epilogue: cooperation, the state and anarchy

freedom, that is, to enjoy one's legal rights of person and property
undisturbed by the encroachments of others. 18 The state must therefore
investigate and settle disputes about such encroachments and punish
transgressions of its laws, since these threaten security. 19 Humboldt
never considers the possibility that disputes could be settled and crimes
punished directly by the people themselves without the help of the state.
Indeed, his only argument in support of the thesis that security must be
provided by the state is that 'it is a condition which man is wholly unable
to realize by his own individual efforts. 20 Yet, if this is true of security,
why is it not also true of other public goods (and perhaps some other
goods too)? A case can of course be made for the special status of
security. One can argue, with Hobbes, that it is fundamental, being a
prerequisite to the attainment of other goods. Humboldt does in fact
take this line : 'Now, without security', he writes, 'it is impossible for man
either to develop his powers, or to enjoy the fruits of so doing.' 21
However, in the first place, it still remains to be shown that security
cannot be realized without the help of the state, and secondly, it can be
argued that if the state is required to provide security, then for the same
reasons it will be required to provide other public goods; in other words,
even when they enjoy state-enforced security, citizens will not necessarily
be able to obtain other things which they want without the further
intervention of the state, which Humboldt expressly forbids.

Nevertheless, the arguments which Godwin uses - and Humboldt
refrains from using - against any sort of state intervention are eloquently
set out by Humboldt in his case against the intervention of the state in
matters not involving security or defence. Here, in particular, is
Humboldt on the effects of the state on altruism and voluntary
cooperation:

As each individual abandons himself to the solicitous aid of the State, |
so, and still more, he abandons to it the fate of his fellow-citizens. This j
weakens sympathy and renders mutual assistance inactive: or, at |
least, the reciprocal interchange of services and benefits will be most I
likely to nourish at its liveliest, where the feeling is most acute that I
such assistance is the only thing to rely upon. 22 |

In Mill's On Liberty we do not encounter this argument until, at the
end of the essay, he considers cases in which the objections to
government interference do not turn upon 'the principle of liberty'.

epilogue: cooperation, the state and anarchy 175

These include cases, he says, in which individuals should be left to act by
themselves, without the help of the state, as a means to their own
development and of 'accustoming them to the comprehension of their
joint interests, the management of joint concerns - habituating them to
act from public or semi-public motives, and guide their conduct by aims
which unite instead of isolating them.' 23 The argument appears also in
the Principles of Political Economy, as 'one of the strongest of the reasons
against the extension of government agency'. 24 Nevertheless, Mill gives
to state interference a considerably wider scope than does Humboldt. In
addition to the maintenance of security, 25 he allows a number of other
important exceptions to his general rule of non-interference. 26

One of these exceptions is of peculiar interest here. The exception
essentially concerns 'free-rider' situations. Mill gives the example of
collective action by workers to reduce their working hours. In such
situations, he says, no individual will find it in his interest to cooperate
voluntarily, and the more numerous are those others who cooperate the
more will he gain by not cooperating; so the assistance of the state is
required to 'afford to every individual a guarantee that his competitors
will pursue the same course, without which he cannot safely adopt it
himself. 27 Penal laws, he goes on to say, are necessary for just this
reason: 'because even an unanimous opinion that a certain line of
conduct is for the general interest, does not always make it people's
individual interest to adhere to that line of conduct'. This is all Mill has
to say on this subject. He is merely providing an argument for an
exception to the general rule of non-interference. He does not appear to
recognize that the same argument would justify state interference in a
vast class of situations. Nor, at the same time, does he appear to
recognize that his general case against the interference of the state could
be applied in all of these situations, including all aspects of the provision
of peace and security.

Rationality

In the last chapter, I criticized Hobbes for drawing the conclusion that
government is the only means whereby men may be coerced to
Cooperate and, more fundamentally, for his relatively static treatment of
the problem. I went on to note that Hume's political theory, while it also
suffers from the first of these failings, to some extent remedies the second :

176 EPILOGUE: COOPERATION, THE STATE AND ANAKi.ni

but although his approach is more dynamical, Hume concurs with
Hobbes in concluding that Cooperation will not occur voluntarily,
neglecting the possibility that the voluntary Cooperation of all in-
dividuals may occur in a dynamic game because the adoption of a
conditionally Cooperative strategy is rational under certain conditions
for each individual. Finally, I questioned the assumption of both Hobbes
and Hume that a government-enforced state of Peace is preferred by
every individual to the state of War, and in this connection I drew
particular attention to the way in which government might not only
impose costs on the individual but in addition diminish the satisfaction
he derives from being in the state of Peace.

This last point refers only to a static effect of government - to an effect
which operates in the same way at each point in time without causing
cumulative changes.

Even when time is explicitly brought into the analysis in the way this is
done in Hume's political theory and in chapters 3 and 4, the resulting
formulation is static in a further sense, namely, that 'human nature' is
taken as given and assumed to be constant. More precisely, egoism or
some combination of egoism and altruism is assumed once and for all to
characterize each individual; it undergoes no modification at any stage
during the 'game', no matter how the players have previously behaved;
and it remains unaltered upon the introduction of government and by
the continued presence of government.

This assumption could be modified, and a further dynamic element
injected, by allowing the combination of egoism and altruism to change
over time, while still assuming that at each point of time an individual
can be characterized by a utility function embodying some combination
of egoism and altruism. In particular, it could now be assumed that the
egoism-altruism combination changes in a way which depends on the
history of the players' choices in previous games and on whether these
choices were made voluntarily or as a result of the presence of state
sanctions.

Modification of this sort would already take us outside the 'abstract
man' framework which I mentioned at the start of this chapter, for it
introduces an individual whose 'human nature' is no longer given and
fixed but is partly determined by his changing social situation (including
the effects on him of the state) and is something which to some extent he
himself creates.

The effects of the state on individual preferences were the subject of the
preceding section. The arguments put forward there were not rigorously
demonstrated and no conclusive evidence was given in their support (I
doubt if this is possible). But even if it is conceded only that they may be
true, it follows that it is not at all clear what can be assumed about
'human nature' at any point in time, in particular what the structure of
preferences would be in the absence of the state. The assumptions made
by Hobbes and Hume were supposed to characterize human behaviour
in the absence of the state; but perhaps they more accurately describe
what human behaviour would be like immediately after the state has
been removed from a society whose members had for a long time lived
under states. This is surely the mental experiment which Hobbes and
Hume were performing.

Although Hobbes spoke in Leviathan of many different characteristics
of individuals, the core of his political theory makes essential use of only
one of these, namely the individual's egoism or some combination of
egoism and negative altruism. The same is true of Hume's political
theory, except that the negative altruism of Hobbes is replaced by a
severely limited positive altruism. I have tentatively suggested in this
chapter that these assumptions tend to be self-fulfilling, in the sense that,
if they were not true before the introduction of the state, which they are
said to make necessary, they would in time become true as a result of the
state's activity, or, if individuals already lacked sufficient positive
altruism to make the state unnecessary, they would 'learn', while they
lived under the care of the state, to possess even less of it.

It has often been argued that the choice of the scope and form of social
institutions (such as the state) must be based on 'pessimistic' assump-
tions, so that they will be 'robust' against the worst possible conditions
(such as a society of egoistic or even negatively altruistic individuals) in
which they might be required to operate. It is assumed in such arguments
that if an institution can 'work' (or work better, in some sense, than the
alternatives) when everyone is, for example, egoistic, then it will certainly
do the same when some or all people are positively altruistic. But if the
institutions themselves affect individual preferences - affect the content
of the assumptions from which their relative desirability has been
deduced - then this approach is inappropriate and may be dangerously
misleading. If there is any truth in the arguments I have been making - if
the state is in part the cause of changes in individual preferences - then

178 epilogue: cooperation, the state and anarchy

we cannot deduce from the structure of these preferences that the state is
desirable. Indeed, it is not even clear in this case what it means to say that
the state is desirable. The same objection can be made to any theory
which seeks to justify or prescribe or recommend an institution, rule,
practice, technology, or any set of arrangements in terms of given and
fixed preferences if these are changed over time by whatever it is that is
being justified.

The theory I have been criticizing and the analysis of cooperation in
chapters 2-4 are founded on what has been called the thin theory of
rationality. This is the account of rationality which is almost universally
taken for granted by economists (and not just neoclassical economists).
On this account, first, rationality is relative to given preferences (or more
generally attitudes) and beliefs, which are assumed to be consistent and
do not change over time, and the agent's actions are instrumental in
achieving the given aims in the light of the given beliefs. Secondly, the
agent is assumed to be a pure egoist. A somewhat less thin account would
admit a measure of altruism (as was done in chapter 5). If the mix of
egoism and altruism, or the propensity to act altruistically, was allowed
to vary over time, then one of the components of the first characteristic of
the thin account would also have been relaxed, but rationality would still
be of the instrumental kind. The third crucial feature of the thin
conception of rationality is that the range of incentives assumed to affect
the agent is limited. As I emphasized in chapter 1, Olson's theory of
collective action limits them to the increase in the public good that
results from the individual's contribution, the resources he expends in
making this contribution and also his contribution to organizational
costs, and selective incentives which themselves are limited to the
'material' and the 'social'. Without a limitation on the range of
incentives, a rational choice theory is liable to become a tautology.

Now I want to emphasize that nothing I have said in this chapter
implies that the thin account of rationality cannot provide a satisfactory
foundation for any kind of theory. In fact, my view is that the explanation
of states of affairs or outcomes (however unintended these may be) in
terms of individual actions, and the explanation of actions in terms of
attitudes and beliefs using a thin account of rationality of some sort, are
indispensable parts of any explanatory social theory. 28 My objection is
to the use of the thin account of rationality in 'evaluative' theories, such

epilogue: cooperation, the state and anarchy 179

as the liberal (or, one might say, the neoclassical) justification of the
state.

So it is not my view that the 'thin' theory of collective action, of which a
theory of conditional cooperation such as that developed in chapters 3
and 4 above would be a part, is an unrealistic, inapplicable theory
because it rests on a thin conception of rationality. (I do nevertheless
believe that the thin theory has much more explanatory power in certain
sorts of situations than in others. I have tried elsewhere to characterize
these situations - as well as to assess the prospects of founding
explanatory theories on alternative conceptions of rationality - and I
will not repeat the arguments here.) 29

Nor do the arguments I have made in this chapter require me to
abandon rational choice explanation - or methodological individualism
more generally - and embrace some version of structuralism instead.
The fact that individual actions, preferences and beliefs are caused - by
states, for example, or by any sort of structure - does not make them
explanatorily irrelevant. Just as individual actions, attitudes and beliefs
are in part the products of and must be partly explained by, amongst
other things, structures, so also are structures - or collective action or the
origin and evolution of states - in part the products of and must be partly
explained by individual actions. 30

Annex: the theory of metagames

I am aware of only one attempt to 'rationalize' Cooperation in the one-
shot Prisoners' Dilemma game. This is the theory of metagames
proposed by Nigel Howard in his Paradoxes of Rationality. 1 He believes
the theory to be predictive: Cooperation in the Prisoners' Dilemma is
'rational' if a player reasons in a certain way, and this mode of reasoning
is claimed to be characteristic of real persons.

If Cooperation is 'rational' in the ordinary game, then it should also be
'rational' in the supergame. Our efforts in chapters 3 and 4 were clearly
unnecessary if Howard's argument is valid. Anatol Rapoport believes
that it is. In an enthusiastic article, he has stated that Howard's theory
has 'resolved' the 'paradox' of the Prisoners' Dilemma, reconciling
individual and collective rationality. 2

In this part of the annex, I state why I believe that Howard and
Rapoport are mistaken. The relevant part of Howard's argument as it
applies to the Prisoners' Dilemma is first briefly presented with reference
to the two-person, two-strategy case which we considered in chapter 3.

Consider the game with the following payoff matrix:

C D

x,x z,y

y, z w, w

where y > x > w > z. Call this now the basic game.

Suppose now that player 2's choices are not between the basic
strategies C and D, but between the conditional strategies (Howard calls
them 'policies') consisting of all the mappings from player l's basic
strategies to his own. Let SJS 2 denote the conditional strategy whereby

180

annex: the theory of metagames

181

player 2 chooses Si if player 1 chooses C and S 2 if he chooses D. Then 2's
conditional strategies are:

C/C: to choose C regardless of player l's choice,
D/D: to choose D regardless of player l's choice,
C/D: to choose the same strategy as player 1,
D/C : to choose the opposite of player l's strategy.

If player 2's choices can in fact be made dependent upon player l's
choices in this way, then it is as if the two are playing in a game whose
payoff matrix is:

C/C

D/D

C/D

D/C

X, X

2, y

X, X

z,y

y, *

|w, w|

w, w

y, z

This is called the 2-metagame. Its only equilibrium is (D, D/D),
corresponding to the only equilibrium (D, D) in the basic game.

Suppose next that player l's choices are not between the basic
strategies C and D but are conditional upon the conditional strategies of
player 2 in the 2-metagame. Let SJS^Si/S^ denote the conditional
strategy whereby player 1 chooses S, if player 2 chooses C/C, S 2 if he
chooses D/D, S 3 if he chooses C/D, and S 4 if he chooses D/C. If the
players' choices can in fact be made interdependent in this way, then it is
as if they are playing a game whose payoff matrix is that shown in table 4.

This game is called the Yl-metagame. It has three equilibria, which are
marked in the payoff matrix; but if the two players were indeed playing
in this game, they would not expect the uncooperative equilibrium
(D/D/D/D, D/D) to occur, for each of them strictly prefers either of the
other two equilibria. Both of these two other equilibria are outcomes of a
single strategy of player 2, and since D/D/C/D weakly dominates
C/D /C/D for player 1, both players should expect (D/D/C/D, C/D) to be
the outcome. It is therefore the outcome. (D/D/C/D, C/D) corresponds to
(C, C) in the basic game. In this way, according to Howard, mutual
cooperation is rationalized even in the one-shot game.

A similar outcome occurs in the '21-metagame', where player l's
strategies are conditional upon the choices of player 2, which are in turn
conditional upon the basic strategies of player 1. That is to say, the
outcome is (C/D, D/D/C/D), corresponding to (C, C) in the basic game.

182 annex: the theory of metagames

Table 4. Payoff matrix for the 12-metagame

c/c

D/D

C/D

D/C

c/c/c/c

X, X

z, y

X, X

z, y

D/D/D/D

y, 2

1 w , w|

w, w

y> z

D/D/D/C

y. z

w, w

z, y

D/D/C/D

y, z

w, w

\x,x\

y> z

D/D/C/C

y, 2

w, w

X, X

z,y

D/C/D/D

y, z

z, y

w, w

y. z

D/C/D/C

y. z

z, y

w, w

z, y

D/C/C/D

y. z

z, y

X, X

y. z

D/C/C/C

y. z

z. y

X, X

z, y

C/D/D/D

X, X

w, w

y, z

cmiDic

X X

w, w

z, y

C/D/C/D

X, X

w, w

\x,x\

y. z

C/D/C/C

X, X

w, w

X, X

z, y

C/C/D/D

X, X

z, y

w, w

y. z

C/C/D/C

X, X

z,y

w, w

z, y

C/C/C/D

X, X

z, y

X, X

y. z

Conditional strategies of a higher order could be considered. Thus
player 2's strategies could be conditional upon those of player 1 in the 12-
metagame, whose payoff matrix is exhibited in table 4. (The resulting
game is called the '2 12-metagame'.) However, neither this nor any higher
order metagame would yield new equilibria, for Howard shows that all
equilibria corresponding to distinct outcomes in the basic game are
revealed in the n ,h -order metagames in which each of the n players is
'named' once and only once - the 12-metagame and the 21 -metagame in
this two-person Prisoners' Dilemma instance.

I have two comments to make on this theory of metagames.

1. The first is for me decisive in rejecting the theory as an explanation
of behaviour in the one-shot Prisoners' Dilemma in which, as I have
assumed, binding agreements are not possible. In this game, the players
choose their strategies independently; they are in effect chosen simul-
taneously, with no knowledge of the other's strategy. And no matter how
much they may indulge in 'metagame reasoning' they must in fact
ultimately choose one of their basic strategies.

In metagame theory, on the other hand, the player's strategies are
required to be interdependent, even in this one-shot Prisoner's Dilemma
game.

In the ordinary game, strategies could be made interdependent by use
of a 'referee', not in the game theorist's usual sense, but in the sense of a

annex: the theory of metagames

183

third party who would be notified of the players' strategies, compare
them, and ensure that a conditional strategy is in fact made dependent
upon the specified strategies of others. However, this is equivalent to the
user of a conditional strategy making his choice of a basic strategy after
the choices of those whose strategies his depends upon.

Furthermore, the referee could in general decide a unique outcome
only if the conditional strategies were of the appropriate orders of
'sophistication'. The 'resolution' of the two-person Prisoners' Dilemma
takes place in the 12- or 21 -metagame. Each of these games is
asymmetric in the sense that one player's strategies are first-order
conditional, while the other's are second-order conditional. This asym-
metry is of course essential to the resolution, for if both players use
conditional strategies of the same order, then some conditional strategy
combinations do not yield determinate outcomes (as when, for example,
each player would Cooperate if and only if the other Cooperates). This
asymmetry is to some extent arbitrary; or rather, it emphasizes again
that one player's choices of basic strategy must in fact follow the other's.

Of course, a player may try to ensure that other players will act
Cooperatively, by announcing his intention to use a conditionally
Cooperative strategy, and generally by bargaining with them. However,
such exchanges, supposing them to be possible, would not have the effect
of producing mutual Cooperation, unless agreements reached in this
way were binding. Such agreements are ruled out in the specification of
the game. In any case, if a mechanism for enforcing agreements existed,
then the players would presumably have had little difficulty in agreeing
on mutual Cooperation in the first place, and there would be no need of a
theory of metagames to explain this.

Howard is neither clear nor consistent about the interpretation to be
placed upon strategies in metagames. He often suggests that metagame
strategies are made interdependent through actual bargaining (as on
p. 101 of Paradoxes of Rationality and in applications of the theory
throughout the book) and that a player's choice follows certain other
players' choices in full knowledge of them (pp. 23, 27, 54 and 61 for
example). Elsewhere, however, he seems to say that the choices are not
actually sequential; the players behave as i/they were. Thus (at the first
order of sophistication), a player (k, say) 'sees' the other players choosing
basic strategies, which he 'correctly predicts', while he himself plays as if
he were in the /c-metagame, his strategies being conditional on the other

184

annex: the theory of metagames

players' basic strategies. Metagames of higher order are reached
('subjectively') by similar reasoning. 3

If the players can negotiate binding agreements or if in some other way
their choices are made interdependently, then they are not playing in the
Prisoners' Dilemma game which I have been discussing in this book. Yet
Howard clearly assumes that they are. Part of his case for the need for a
theory such as his ownls based on the 'breakdown of rationality' (as this
concept is used in conventional game theory) which is indicated,
according to Howard, by the standard analysis of the Prisoners'
Dilemma one-shot game.

The conclusions of the standard analysis of this game may be
distressing; but they are unaffected by a consideration of metagames.

2. My second comment is that, if bargaining or any other dynamic
process is indeed the object of study, then the one-shot game (with or
without its metagames) is in any case an inappropriate model. In
bargaining, there are sequences of choices; there are bluffs, threats and
promises; there is learning and adaptation of expectations; the value of
an outcome is discounted with future time; and so on. These things are
not explicitly taken into account in the theory of metagames.

Notes

1. Introduction: the problem of collective action

1 This is a caricature of Hobbes's argument. In chapter 6, 1 give a more detailed
account, making use of ideas developed in chapters 3-5.

2 Both of these are true of William J. Baumol's Welfare Economics and the
Theory of the State, second edition (London: G. Bell, 1965). Much of this
book is devoted to the failure of individuals to provide themselves
voluntarily with public goods, but I think it is fair to say that 'the Theory of
the State' is missing. He is careful to say that, before it is concluded that state
action to ensure the supply of public goods is justified, all the costs of state
action must also be taken into account (p. 22 in the introduction added to the
second edition); nevertheless there is a presumption that only the state could
ensure this supply.

3 Two examples are William Ophuls, 'Leviathan or oblivion?', in Herman E.
Daly (ed.), Toward a Steady-State Economy (San Francisco: W. H. Freeman,
1973), and Robert L. Heilbroner, 'The human prospect', The New York
Review of Books, 24 January 1974.

4 An approximate example is A Blueprint for Survival, by the editors of The
Ecologist (Harmondsworth, Middlesex: Penguin Books, 1972; originally
published as Vol. 2, No. 1 of The Ecologist, 1972). Their goal is not wholly
anarchist, but it does include 'decentralisation of polity and economy at all
levels, and the formation of communities small enough to be reasonably self-
supporting and self-regulating'. For an anarchist's account of the necessity of
anarchist society on ecological grounds, see Murray Bookchin, 'Ecology and
Revolutionary Thought', in Post-Scarcity Anarchism (Berkeley, California:
The Ramparts Press, 1971).

5 See, for example, Anthony Crosland, A Social Democratic Britain (Fabian
Tract no. 404, London, 1971), and Jeremy Bray, The Politics of the
Environment (Fabian Tract no. 412, London, 1972).

6 Garrett Hardin, 'The tragedy of the commons', Science, 162 (13 December
1968), 1243-8.

7 For a brief account of the overexploitation of whales and various species of
fish, see Paul R. Ehrlich and Anne H. Ehrlich, Population, Resources,
Environment, second edition (San Francisco: W. H. Freeman, 1972), pp.

185

186

NOTES TO PAGES 6-14

125-34. See also Frances T. Christy and Anthony Scott, The Common Wealth
in Ocean Fisheries (Baltimore: Johns Hopkins Press, 1965).

8 The word 'consumption' should perhaps be used only in connection with
private goods, where it has a clear meaning. To speak of 'consuming' national
defence, wilderness and radio broadcasts is somewhat strained, but for want
of a suitable word to cover a variety of applications, I follow the custom of the
economists and retain the word. In many cases, 'consume' means 'use'. Cf.
Jean-Claude Milleron, 'Theory of value with public goods: a survey article',
Journal of Economic Theory, 5 (1972), 419-77, at pp. 422-3.

9 This follows Samuelson's most recent usage (though I have added the
requirement that a public good be also non-excludable). Samuelson had
defined a public good as one which was consumed equally by every individual,
so that x 1 =x 2 =. . . = x, where x' is the i' lh individual's consumption of the
good and x is the total amount available ; and he defined a private good as one
which could be divided amongst individuals so that x 1 +x 2 +. . . = x. See
Paul A. Samuelson, 'The pure theory of public expenditure', Review of
Economics and Statistics, 36 (1954), 387-9. In his 1955 paper, he admitted
that these were two pure, polar cases; and most recently he has abandoned
these two poles in favour of a 'knife-edge pole' of the pure private good and
'all the rest of the world in the public good domain'. Samuelson, 'Diagram-
matic exposition of a theory of public expenditure', Review of Economics and
Statistics, 37 (1955), 350-6; and 'Pure theory of public expenditure and
taxation', in J. Margolis and H. Guitton (eds), Public Economics (London:
Macmillan, 1969).

10 Cf. William Loehr and Todd Sandler (eds), Public Goods and Public Policy
(Beverly Hills: Sage, 1978), p. 2 and ch. 6.

1 1 For a fuller discussion of social order as a public good, see my Community,
Anarchy and Liberty (Cambridge: Cambridge University Press, 1982),
sections 2.1 and 2.3.

12 Mancur Olson, The Logic of Collective Action (Cambridge, Mass. : Harvard
University Press, 1965), p. 36.

13 Olson, The Logic, p. 48.

14 Cf. The Logic, p. 50, note 70.

15 Russell Hardin, Collective Action (Baltimore: The Johns Hopkins Press for
Resources for the Future, 1982), pp. 41-2.

16 Including Olson himself, as we shall see when we come to discuss altruism in
chapter 5.

17 Hardin, Collective Action, p. 44.

18 This qualifies the very useful treatment of this issue in Collective Action, ch. 3.

19 Olson, The Logic, p. 29, note 46.

20 The Logic, p. 132.

21 The Logic, p. 61, note 17. But see also p. 160, note 91.

22 On these non-instrumental motivations, see my 'Rationality and re-
volutionary collective action', in Michael Taylor (ed.), Rationality and
Revolution (Cambridge: Cambridge University Press, 1987).

23 The story about two prisoners, which gave the game its name, can be found in

NOTES TO PAGES 16-28

187

R. Duncan Luce and Howard Raiffa, Games and Decisions (New York: John
Wiley, 1957), p. 95.

24 Russell Hardin, 'Collective action as an agreeable n-Prisoners' Dilemma',
Behavioral Science, 16 (1971), 472-81.

25 Jon Elster, 'Rationality, morality, and collective action', Ethics, 96 (1985),
136-55. The weak definition, identifying collective action problems with the
Prisoners' Dilemma, is adopted by Elster in 'Weakness of will and the free-
rider problem', Economics and Philosophy, 1 (1985), 231-65 - but then again
he admits that 'it does not . . . cover all the cases that intuitively we think of as
collective action problems'.

26 Jon Elster, 'Some conceptual problems in political theory', in Brian Barry
(ed.), Power and Political Theory (London: Wiley, 1976), at pp. 248-9.

27 Colin Clark, 'The economics of overexploitation', Science, 181 (17 August
1973), 630^.

28 Taylor, Community, Anarchy and Liberty.

29 The two points in this paragraph where made by Brian Barry in Sociologists,
Economists and Democracy (London: Collier-Macmillan, 1970) at pp. 27-39.

30 Olson, The Logic, Appendix added in 1971, p. 177.

31 Samuel L. Popkin, The Rational Peasant: The Political Economy of Rural
Society in Vietnam (Berkeley: University of California Press, 1979), espe-
cially ch. 3.

32 Robert McC. Netting, Balancing on an Alp: Ecological Change in a Swiss
Mountain Community (Cambridge: Cambridge University Press, 1981),
especially ch. 3.

33 Lester Brown and Edward Wolf, Soil Erosion: Quiet Crisis in the World
Economy (Washington, D.C. : Worldwatch Institute, 1984). According to this
report, U.S. farms are losing topsoil at the rate of 1.7 billion tonnes a year.
The New York Times (10 December 1985) reports that the U.S. Congress
looks set to vote to pay farmers to stop farming up to 40 million acres of the
worst affected land.

34 See, for example, Michael H. Glantz (ed.), Desertification: Environmental
Degradation In And Around Arid Lands (Boulder, Colorado : Westview Press,
1977).

35 The following comments, which are critical of the property rights school's
treatment of the 'tragedy of the commons', do not imply a wholesale rejection
on my part of the property rights approach.

36 A. A. Alchian and Harold Demsetz, 'The property rights paradigm', Journal
of Economic History, 33 (1973), 16-27.

37 See S. V. Ciriacy-Wantrup and Richard C. Bishop, ' "Common property" as
a concept in natural resources policy', Natural Resources Journal, 15 (1975),
713-27.

38 See, for example, Harold Demsetz, 'Toward a theory of property rights',
American Economic Review (Papers and Proceedings), 57 (1967), 347-59.

39 Carl J. Dahlman, The Open Field System and Beyond (Cambridge: Cam-
bridge University Press, 1980).

40 See, for example, Eirik G. Furobotn and Svetozar Pejovich, 'Property rights

188

NOTES TO PAGES 29-49

and economic theory : a survey of recent literature', Journal of Economic
Literature, 10 (1972), 1137-62.

41 Edna Ullman-Margalit, The Emergence of Norms (Oxford : Clarendon Press,
1977). A 'generalized PD-structured situation ... is one in which the
dilemma faced by the . . . participants is recurrent, or even continuous'
(p. 24); but Ullman-Margalit gives no analysis of iterated games or takes any
account of their distinctive problems (so, amongst other things, does not see
that cooperation in these situations can occur without norms enforced by
sanctions). Incidentally, very little of this book actually deals with the
emergence of norms ; it is mainly taken up with generally informal discussion
of some very simple games.

42 The Emergence of Norms, pp. 22 and 28 ; my emphasis.

2. The Prisoner's Dilemma, Chicken and other games in the provision of public
goods

1 This chapter draws on Michael Taylor and Hugh Ward, 'Chickens, whales,
and lumpy goods: alternative models of public goods provision', Political
Studies, 30 (1982), 350-70.

2 Russell Hardin, Collective Action, p. 25.

3 Russell Hardin, 'Collective action as an agreeable n-Prisoners' Dilemma',
Behavioral Science, 16 (1971), 472-81; and again, virtually unchanged, in
Collective Action, ch. 2.

4 Whether or not this is true of the Cournot analysis, to which one brief section
is devoted, is not very important, since the Cournot approach is of little use
anyway. In my view, what is involved is only a sort of pseudo-dynamics; no
actual process is described.

5 See Dennis C. Mueller, Public Choice (Cambridge: Cambridge University
Press, 1979), ch. 2.

6 An argument to this effect is made in Taylor and Ward, 'Chickens, whales
and lumpy goods'.

7 c and b are here assumed to be independent of N. This is a reasonable
assumption in many cases, including the voting one discussed earlier, but not
in all cases.

8 This analysis is based on that given in Taylor and Ward, 'Chickens, whales,
and lumpy goods'. It can also be found in Amnon Rapoport, 'Provision of
public goods and the MCS paradigm', American Political Science Review, 79
(1985), 148-55. Rapoport also considers 'heterogeneous' cases where player i
does not take the other players to be equally likely to Cooperate.

9 Compare Rapoport, 'Provision of public goods', pp. 150-1.

10 For further details, see Taylor and Ward, 'Chickens, whales, and lumpy
goods', pp. 368-70.

1 1 See Hugh Ward, 'The risks of a reputation for toughness : strategy in public
goods provision problems modelled by Chicken supergames', British Journal
of Political Science, 17 (1987).

NOTES TO PAGES 50-62

189

12 My presentation here draws on James M. Buchanan, The Demand and
Supply of Public Goods (Chicago: Rand McNally, 1968); Buchanan,
'Cooperation and conflict in public-goods interaction', Western Economic
Journal, 5 (1967), 109-21 ; Gerald H. Kramer and Joseph Hertzberg, 'Formal
theory', in volume 7 of The Handbook of Political Science, F. Greenstein and
N. Polsy, eds. (Reading, Mass.: Addison- Wesley, 1975); and Taylor and
Ward, 'Chickens, whales, and lumpy goods'.

13 If the public good is inferior, then when the Others provide an additional unit
of it the individual will reduce his provision of it by more than one unit.

14 John Chamberlin, 'Provision of collective goods as a function of group size',
American Political Science Review, 68 (1974), 707-13; Martin C. McGuire,
'Group size, group homogeneity and the aggregate provision of a pure public
good under Cournot behavior', Public Choice, 18 (1974), 107-26.

15 Chamberlin argues that if there is perfect rivalness (i.e., the good 'exhibits the
same rivalness of consumption as does a private good') but non-
excludability, then total production at equilibrium necessarily decreases as N
increases. He correctly points out that if we abandon the assumption of
perfect nonrivalness, the reaction curves vary with N. But they need not vary
in the particular manner he assumes. Changes in an individual's reaction
curve as N varies can come about as a result of changes in the transformation
function facing the individual or in his indifference map. This follows from
the remarks made at the end of the last section. Chamberlin, like many
others, conflates indivisibility and nonrivalness, but my point that the
reaction curves need not vary in the way he assumes holds whether their
variation as JV varies is due to changes in the transformation function or the
indifference map or both. The (true) statement that when the public good is
not purely indivisible or there is some degree of rivalness, the group's total
production may increase or decrease, is said by Chamberlin to hold only for
the cases 'intermediate' between perfect nonrivalness and perfect rivalness;
but here too the two patterns of variation in the reaction curves as N varies
which Chamberlin considers do not exhaust the possibilities. In any case,
reaction curves will be radically different from the ones he considers in some
important cases, in particular in lumpy goods cases, as discussion in the text
below suggests.

3. The two-person Prisoners' Dilemma supergame

1 The number r, such that a, = 1/(1 +r,) is sometimes called the rate of time
preference.

2 This has not deterred some economists and game theorists in recent years
from studying finite supergames and infinite supergames with no discount-
ing. On infinite games without discounting, see especially A. Rubinstein,
'Equilibrium in supergames with the overtaking criterion', Journal of
Economic Theory, 21 (1979), 1-9; Steve Smale, 'The Prisoners' Dilemma and

190

NOTES TO PAGES 62-75

dynamical systems associated to non-cooperative games', Econometrica, 48
(1980), 1617-34; and Robert J. Aumann, 'Survey of repeated games', pp.
11-42 in R. J. Aumann, et ai, Essays in Game Theory and Mathematical
Economics in Honor of Oskar Morgenstern (Mannheim/Wien/Zurich: Bib-
liographisches Institut, 1981). For finitely repeated games, see especially the
works cited in the next note. I think, however, that most economists believe
that supergames of indefinite length with discounting are generally the most
appropriate model.

3 This well-known result for the finitely repeated Prisoners' Dilemma no
longer holds if a small amount of uncertainty is introduced into the game. If
players are not quite certain about each other's motivations, options or
payoffs, the backwards induction argument cannot be applied. See David M.
Kreps, et ai, 'Rational cooperation in the finitely repeated Prisoners'
Dilemma', Journal of Economic Theory, 27 (1982), 245-52; David M. Kreps
and Robert Wilson, 'Reputation and imperfect information', Journal of
Economic Theory, 27 (1982), 253-79; and Kreps and Wilson, 'Sequential
equilibria', Econometrica, 50 (1982), 863-94.

4 Mixed strategies are ruled out, chiefly because they do not seem to
correspond to any realistic course of action in the real world problems of
public goods provision which are of interest in this book.

5 This chapter extends, in certain respects, earlier work on the two-person
Prisoners' Dilemma supergame in Martin Shubik, Strategy and Market
Structure: Competition, Oligopoly, and the Theory of Games (New York:
Wiley, 1959); Shubik, 'Game theory, behavior, and the paradox of the
Prisoners' Dilemma: three solutions', Journal of Conflict Resolution, 14
(1970), 181-93; and Michael Nicholson, Oligopoly and Conflict (Liverpool:
Liverpool University Press, 1972).

6 Observe that the payoff matrix is symmetric: it remains unchanged when the
players are interchanged (relabelled). Abandoning symmetry (while retain-
ing the Prisoners' Dilemma ordering of the payoffs) would require modifi-
cations of detail (in the conditions below for strategy vectors to be equilibria,
the payoffs would have to be subscripted as well as the discount factors); but
the general argument would not be changed. In this book, I wish to isolate
the Prisoners' Dilemma element.

7 Robert Axelrod, The Evolution of Cooperation (New York: Basic Books,
1984), pp. 208 9.

8 The Evolution of Cooperation, p. 173.

9 This last point is made by Norman Schofield, 'Anarchy, altruism and
cooperation', Socio/ Choice and Welfare, 2 (1985), 207-19.

10 The Evolution of Cooperation, p. 11 and p. 216 note 3.

11 Russell Hardin, Collective Action, p. 171.

12 Collective Action, p. 171.

13 Martin Shubik, 'Game theory, behavior, and the paradox of the Prisoner's
Dilemma: three solutions', Journal of Conflict Resolution, 14 (1970), 181-93.

14 This strategy was introduced, I think, by Martin Shubik in Strategy and

NOTES TO PAGES 75-84

191

Market Structure, at pp. 224-5. Its analogue for any noncooperative game is
studied by James W. Friedman in 'A non-cooperative equilibrium for
supergames', The Review of Economic Studies, 38 (1971), 1-12. See also John
McMillan, 'Individual incentives in the supply of public inputs', Journal of
Public Economics, 12 (1979), 87-98.

15 Shubik, 'Game theory, behavior, and the paradox of the Prisoner's
Dilemma'.

16 A parenthetical comment is appropriate here on the condition 2x> y + z,
which is stipulatively required in some accounts of the Prisoners' Dilemma,
on the grounds that, if it did not hold, then alternating between (C, D) and (D,
C) would be preferable to mutual Cooperation. (It is required, for example,
by Anatol Rapoport and Albert C. Chammah in their pioneering book, The
Prisoner's Dilemma, Ann Arbor: The University of Michigan Press, 1965, p.
34.) The condition does indeed rule this out in a Prisoners' Dilemma
supergame without discounting (and accordingly plays an important role in
models of this supergame such as the one discussed in the annex); but in the
present analysis, in which players discount future payoffs, it is not sufficient to
make either of the alternation patterns Pareto-preferred to mutual Cooper-
ation. In fact, for the B' player to prefer mutual Cooperation throughout the
supergame to (B, B') or (B', B) we require a, > (y-x)/(x-z), and for the B
player to prefer it we require a f < (x — z)j(y — x). These two inequalities, then,
each holding for both players, are the necessary and sufficient conditions for
mutual Cooperation to be preferred by both players to either of the
alternation outcomes. I have preferred not to stipulate this but instead to
analyse the conditions under which alternation occurs. The condition
2x > y + z is nevertheless a necessary condition for (B, B) to be preferred to
(B, B') and (B\ B) by the B' player (and is therefore a necessary condition for
(B, B) to be an equilibrium), since one of the necessary conditions for this is
a ( > (y — x)/(x — z), and since a { < 1 we must have (y - x)/(x — z)< 1, that is,
2x > y + x.

4. The N-person Prisoners' Dilemma supergame

1 This was not made clear in Anarchy and Cooperation.

2 A different definition of the JV-person Prisoners' Dilemma is given by Thomas
C. Schelling in 'Hockey helmets, concealed weapons, and daylight saving: a
study of binary choices with externalities', Journal of Conflict Resolution, 17
(1973), 381-428. He defines a 'uniform multiperson prisoner's dilemma' as a
game such that : (1 ) each player has just two strategies available to him and the
payoffs can be characterized (in effect ) by two functions f(v) and g(v), which are
the same for every individual; (2) every player has a dominant strategy (£>); (3)
f(v) and g(v) are monotonically increasing; (4) there is a number k > 1, such
that if k or more players choose C and the rest do not, those who choose C are

192

NOTES TO PAGES 93-109

better off than if they had all chosen D, but if they number less than k, this is not
true. Schilling's (1) and (2) are also part of my definition. His (3) is a much
stronger requirement than my (iii). And his (4) is stronger than my (ii). For all I
require in (ii) is that the first part of Schelling's (4) holds for k = N; and I leave
open the question of whether fewer than N individuals obtain a higher payoff
when they all Cooperate (and the rest do not) than when they all Defect.
Schelling's definition is therefore more restrictive than mine. His requirement
(4), it seems to me, partly removes the 'dilemma' in the Prisoners' Dilemma.

Other ways of defining the N-person Prisoners' Dilemma more restrictively
than I define it here can be found in Henry Hamburger, 'N-person Prisoner's
Dilemma', Journal of Mathematical Sociology, 3 (1973), 27-48.

3 If there is some symmetry between the payoff functions of different players, the
number of these equilibria might be smaller; but as long as some remain the
problem to be discussed in the text would still arise. Only exceptionally would
the asymmetry be such that there is just one subset of players such that
(B„/C K /D°°) is an equilibrium.

4 Michael Laver, 'Political solutions to the collective action problem', Political
Studies, 28 (1980), 195-209; and Iain McLean, 'The social contract and the
Prisoner's Dilemma supergame', Political Studies, 29 (1981), 339-51.

5 Early indications from simple computer simulation exercises with one such
model (in unpublished work by Hugh Ward) suggest that a pre-commitment
'scramble' could occur which levelled out and stabilized at the desired
subgroup size.

6 See the references in notes 14 and 15 to chapter 3.

7 A very small group may be 'privileged' in Olson's sense (i.e., there is at least one
individual who is willing to provide some of the public good unilaterally), in
which case, as we saw in chapter 2, preferences at any point in time are not
those of a Prisoners' Dilemma, and the whole argument of this chapter is
inapplicable. If there are several such individuals, each of whom has a very
strong interest in the public good, a different problem of strategic interaction
arises. See the discussion in chapter 2.

8 See the brief discussion in chapter 1 and for a fuller account see my
Community, Anarchy and Liberty (Cambridge: Cambridge University
Press, 1982), ch. 2.

9 This conclusion finds some support in Michael Nicholson's work, although
his analysis cannot be compared directly with the one carried out here. See
Oligopoly and Conflict (Liverpool: Liverpool University Press, 1972), Section
3.2. See also his discussion of this type of flexibility in chapter 6.

5. Altruism and superiority

1 Olson, The Logic of Collective Action, p. 64.

2 Brian Barry pointed this out in Sociologists, Economists and Democracy, at
p. 32.

NOTES TO PAGES 111-26

193

3 I have made this argument in more detail in 'Rationality and revolutionary
collective action'.

4 A parenthetical comment is in order to explain why I have made no use of the
model of altruistic behaviour proposed by Howard Margolis in his Selfishness,
Altruism, and Rationality (Cambridge: Cambridge University Press, 1982),
which on the face of it offers a much more realistic account of altruism than the
simple one used here. Certainly, I believe that Margolis's theory is the most
interesting attempt to date to incorporate altruistic motivation into a model of
individual choice; but unfortunately it is radically incomplete and, as far as I
can see, unusable. I have set out my reasons for reaching this conclusion
elsewhere (Ethics, 94 (1983), 150-2) and will not repeat them here. To
summarize drastically, an individual on Margolis's account allocates his
resources between 'selfish' and 'group' interests in such a way as to feel that he
has done his 'fair share', and the rule which yields allocations answering to this
is as follows : 'the larger the share of my resources I have spent unselfishly, the
more weight I give to my selfish interests in allocating marginal resources. On
the other hand, the larger benefit I can confer on the group compared with the
benefit from spending marginal resources on myself, the more I will tend to act
unselfishly' (Margolis, p. 36). So the weight the individual gives to his selfish
interests is a function of the history of his past (altruistic and/or egoistic)
behaviour. But how this weight varies with the individual's history is not
specified. Margolis does not in fact consider in detail any dynamic examples
and it is not at all clear how the model can be applied to dynamic games in
which there is strategic interaction over time.

5 What I have called Games of Difference have been considered by James R.
Emshoff in 'A computer simulation model of the Prisoner's Dilemma',
Behavioral Science, 15 (1970), 304-17. He refers to A,- as the 'competitiveness
parameter'. Pure Difference Games have been studied by Martin Shubik,
'Games of status', Behavioral Science, 16 (1971), 117-29, who calls them
'difference games'. He considers also a further transformation to what he calls
'games of status', in which there are only three different payoffs: one for
winning (when the payoff difference is positive), one for losing (when the
difference is negative) and one for drawing.

6 Sophisticated altruism or something like it is discussed under different names
by Stefan Valavanis, 'The resolution of conflict when utilities interact', Journal
of Conflict Resolution, 2 (1958), 156-69 and Thomas C. Schelling, 'Game
theory and the study of ethical systems', Journal of Conflict Resolution, 12
(1968), 34-44.

6. The state

1 References to Leviathan (abbreviated Lev) are to the pages of the edition by
W. G. Pogson Smith (Oxford: The Clarendon Press, 1909).

194

NOTES TO PAGES 129-66

2 I must thank Brian Barry for helping me to see Leviathan in a more 'dynamic'
light.

3 Alasdair Maclntyre, A Short History of Ethics (London: Routledge and
Kegan Paul, 1967), p. 138.

4 H. L. A. Hart, The Concept of Law (Oxford: The Clarendon Press, 1961),
p. 193.

5 Brian Barry, 'Warrender and his critics', Philosophy, 48 (1968), 117-37, at
p. 125.

6 On symmetry, see chapter 3, note 6.

7 C. B. Macpherson, The Political Theory of Possessive Individualism: Hobbes
to Locke (Oxford: The Clarendon Press, 1962).

8 The citations of Hume give the page numbers of the Selby-Bigge editions : L.
A. Selby-Bigge (ed.), A Treatise of Human Nature (Oxford: The Clarendon
Press, 1888) and Enquiries Concerning the Understanding and Concerning the
Principles of Morals (Oxford: The Clarendon Press, second edition, 1902).
The Treatise is abbreviated to Tr and Enquiry refers to An Enquiry
Concerning the Principles of Morals.

9 David Lewis, Convention: A Philosophical Study (Cambridge, Mass.:
Harvard University Press, 1969), p. 42. Lewis later refines this definition by
adding the condition that it is 'common knowledge' in p that (1), (2) and (3)
obtain. He also considers degrees of convention. But this 'first, rough
definition' will suffice for my purposes.

10 Governments are in fact very active in establishing and modifying conven-
tions and in many cases they make laws of them and punish non-conformists.
If they are pure conventions, this is not necessary. For example, driving on
the 'right' side of the road is almost a pure convention, and once it is
established, there is almost no need for government enforcement : very few
individuals will want to drive on the 'wrong' side. Of course, a central
coordinating agency may be useful in establishing a convention more quickly
and less painfully than it would establish itself 'spontaneously'. But this is not
an argument in favour of government; for such an agency need have no
power, and it need only be ad hoc and temporary: there is no need, in this
connection, for a single agency to take charge of all conventions, and once a
convention is established, the agency in question can be disbanded.

7. Epilogue: cooperation, the state and anarchy

1 Arthur Lehning (ed.), Michael Bakunin: Selected Writings (London: Jonat-
han Cape, 1973; New York: Grove Press, 1973).

2 For introductory accounts of problems of arms races and disarmament in
terms of Prisoners' Dilemma games and of other international interactions in
terms of Chicken games, see Anatol Rapoport, Strategy and Conscience (New
York : Harper and Row, 1964); Glenn H. Snyder, ' "Prisoners' Dilemma" and

NOTES TO PAGES 166-72

195

"Chicken" models in international polities', International Studies Quarterly,
15 (1971), 66-103; Glenn H. Snyder and Paul Diesing, Conflict Among
Nations: Bargaining, Decision Making, and System Structure in International
Crises (Princeton, N.J.: Princeton University Press, 1977); Robert Jervis,
'Cooperation under the security dilemma', World Politics, 30 (1978),
167-214.

3 See, most recently, Kenneth Oye (ed.), Cooperation Under Anarchy (Prince-
ton, N.J.: Princeton University Press, 1986).

4 See the brief account of community in the penultimate section of chapter 1
above.

5 For example, Rupert Emerson, writing on the new nations of Africa in a
volume on Nation-Building, has this to say: 'At the extremes, tribalism can
be dealt with in two fashions - either use of the tribes as the building blocks of
the nation or eradication of them by a single national solidarity. It is the
latter course which is more generally followed.' And William Foltz, speaking
generally of the new nations in his conclusion to this volume, writes : 'The old
argument over the priority of state or nation is being resolved by these
countries' leaders in favour of first building the state as an instrument to
bring about the nation'. See Karl W. Deutsch and William J. Foltz (eds),
Nation-Building (New York: Atherton Press, 1963). On European states, see
for example Charles Tilly, 'Reflections on the history of European state-
making', in Tilly (ed.), The Formation of National States in Western Europe
(Princeton, N.J. : Princeton University Press, 1975), especially at pp. 2 1 -4, 37
and 71.

6 Peter Kropotkin, Mutual Aid: A Factor of Evolution (London: Allen Lane
The Penguin Press, 1972; reprinted from the edition of 1914), p. 197.

7 Richard Sennett, The Uses of Disorder: Personal Identity and City Life
(London : Allen Lane The Penguin Press, New York : Alfred A. Knopf, 1 97 1 ).
This quotation is from the Pelican edition (Harmondsworth, Middlesex:
Penguin Books, 1973), pp. 132-3, by courtesy of Penguin Books Ltd and
Alfred A. Knopf Inc.

8 Richard M. Titmuss, The Gift Relationship : From Human Blood to Social
Policy (London: George Allen and Unwin, New York: Random House,
1970). References here are to the Pelican edition (Harmondsworth, Mid-
dlesex: Penguin Books, 1973), quoted by courtesy of George Allen & Unwin
Ltd and Pantheon Books, a Division of Random House, Inc.

9 The Gift Relationship, pp. 84-5.

10 The Gift Relationship, pp. 256 8.

1 1 Peter Singer, 'Altruism and Commerce : A defense of Titmuss against Arrow',
Philosophy and Public Affairs, 2 (1973), 312-20.

12 This experiment is reported in J. H. Bryant and M. A. Test, 'Models and
helping: naturalistic studies in aiding behavior', Journal of Personality and
Social Psychology, 6 (1967), 400 7. The best source for reports of experiments
of this kind is J. Macaulay and L. Berkowitz (eds), Altruism and Helping
Behavior (New York: Academic Press, 1970), especially the chapters of the

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